Week 14: 20 Apr, |
21 Apr, | 22 Apr, | 23 Apr, | 24 Apr, |

Week 13: 13 Apr, |
14 Apr, | 15 Apr, | 16 Apr, | 17 Apr, |

Week 12: 06 Apr, |
07 Apr, | 08 Apr, | 09 Apr, | 10 Apr, |

Week 11: 30 Mar, |
31 Mar, | 01 Apr, | 02 Apr, | 03 Apr, |

Week 10: 23 Mar, |
24 Mar, | 25 Mar, | 26 Mar, | 27 Mar, |

Week 9: 09 Mar, |
10 Mar, | 11 Mar, | 12 Mar, | 13 Mar, |

Week 8: 02 Mar, |
03 Mar, | 04 Mar, | 05 Mar, | 06 Mar, |

Week 7: 23 Feb, |
24 Feb, | 25 Feb, | 26 Feb, | 27 Feb, |

Week 6: 16 Feb, |
17 Feb, | 18 Feb, | 19 Feb, | 20 Feb, |

Week 5: 09 Feb, |
10 Feb, | 11 Feb, | 12 Feb, | 13 Feb, |

Week 4: 03 Jan, |
03 Feb, | 04 Feb, | 05 Feb, | 06 Feb, |

Week 3: 26 Jan, |
27 Jan, | 28 Jan, | 29 Jan, | 30 Jan, |

Week 2: 20 Jan, |
21 Jan, | 22 Jan, | 23 Jan, | |

Week 1: 25 Aug, |
13 Jan, | 14 Jan, | 15 Jan. | 16 Jan. |

Lecture: Intro to stability theory for autonomous systems. Equilibria. Stability. Instability. Asymptotic stability.

Exercise solutions: ch7 and ch8.

Lecture 9.1: Classification of equilibria for u'=Au when det(A) is not zero, for the 2x2 case.

Spiral, saddle, center, node. Linearization theory. Jacobian.

Transfer of stability: Re(lambda)<0 ==> asym. stability. Unstable solutions.

Nonlinear stability theory: when the linearized classification and stability transfers to the nonlinear system.

Sample exam 3: Some solutions.

Lecture: Sample exam 3 solutions to problems 1,2.

Lecture 9.2: More on stability, linear and almost linear systems, Jacobian, stability of linear systems, stability of almost linear [nonlinear] systems, phase diagrams, classification of nonlinear systems.

Final exam review started. Cover ch5 mostly, some of ch10. Packet distributed on the web.

Final exam details: Less contact with ch3, ch4, ch6 due to extra chapters 8,9 on the final, as compared to the S2008 final exam. Not much change for ch5, ch7, ch10. The new spin is only additional methods for solving DE, especially exp(At) and Laplace resolvent for systems.

Lecture 9.3: Nonlinear stability, phase diagrams, classification. Predator-Prey systems. How to tell which is the predator and which is the prey. Calculations for equilibrium points, linearization, classification of equilibria, impact on the phase diagram. Some sample code for using DEtools and DEplot in maple to make phase diagrams. Exercises 9.1, 9.2.

Exam 3 at 7am and 10:30am.

Final exam review continued. Ch10 Laplace problems. Some ch8 and ch9 problems.

Lecture 9.4: Nonlinear mechanical systems. Hard and soft springs. Nonlinear pendulum. Undamped pendulum. Damped pendulum. Phase diagrams. Energy conservation laws and separatrices.

Slides on Dynamical Systems

Systems theory and examples, 800k. (pdf manuscript)

Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation. (pdf slides)

Introduction to dynamical systems (pdf slides)

Phase Portraits for dynamical systems (pdf slides)

Stability for dynamical systems (pdf slides)

Review and Drill: Electric circuits.

Impedance Z. Reactance S=omega L - (1/C)(1/omega)

Time lag delta/omega. Steady-state periodic solution. Transient solution.

Review: Electrical-mechanical analogy. Faraday's law, Ohm's law, Coulomb's law, Kirchhoff's laws. Voltage drop formulas.

Resonance equations.

mx''+cx'+kx=F0 cos(omega t), omega = sqrt(k/m - c^2/(2m^2)) LQ''+RQ'+Q/C=E0 sin(omega t), LI''+RI'+I/C=omega E0 cos(omega t), omega = 1/sqrt(LC)

Network modeling: Number of independent differential equations used to describe a multi-loop circuit.

Lecture: 7.1, 7.2 How to convert mx''+cx'+kx=F0 cos (omega t) into a dynamical system u'=Au+F(t). Electrical systems u'=Au+E(t) from LRC circuit equations. Electrical systems of order two, mechanical systems of order two, u''=Au+F

Lecture: Second order systems. Examples are railway cars, earthquakes, vibrations of multi- component systems, electrical networks.

Systems theory and examples, 800k. (pdf manuscript)

Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation. (pdf slides)

Lecture: Packages D and P in the diagonalization equation AP=PD, which is equivalent to Fourier's model. In the 2x2 case, Fourier's Model is A(c1 v1 + c2 v2) = c1(lambda1 v1) + c2(lambda2 v2).

Given P and D, find A in the relation AP=PD.

Problem: Given Fourier's model, find A.

Problem: Given A, find Fourier's model.

Problem: Given A, find all eigenepairs.

Problem: Given A, find packages P and D such that AP=PD.

Problem: Give an example of a matrix A which has no Fourier's model.

Problem: Give an example of a matrix A which is not diagonalizable.

Problem: Given 2 eigenpairs, find the 2x2 matrix A.

Lecture: Problems 7.3, 7.4.

Main theorem (ch7, 7.3) on solving u'=Au by eigenanalysis.

Main theorem for Cayley-Hamilton method.

Main theorem for solving u''=Au+F

Main theorem for the exponential matrix.

Methods to solve dynamical systems like x'=x-5y, y'=x-y, x(0)=1, y(0)=2. Cayley-Hamilton method. Laplace resolvent. Eigenanalysis method. Exponential matrix using maple Putzer's method Spectral methods [ch8]

Lecture: Eiegnanalysis examples. How to find Fourier's model. Equivalence of Fourier's model and diagonalization. Matrices P,D and how to find them.

Examples of size 2x2, 3x3, 4x4 that have no Fourier model, that are not diagonalizable.

Examples of size 2x2, 3x3, 4x4 that have a Fourier model, that are diagonalizable.

Computing A from its fourier model.

Computing A from its eigenpairs.

Computing A from P and D.

Slides on applications.

Systems theory and examples, 800k. (pdf manuscript)

Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation. (pdf slides)

Topics:

Brine tanks revisited. Cascades. Re-circulating tanks. Pond pollution. Home heating. Railway cars. Earthquakes.

Detection of model properties from the solutions.

How to tell which methods might have been used to obtain the solution. Solution atoms and characteristic equations.

Separating homogeneous and nonhomogeneous solutions in u=uh+up. Discussion of the various methods for solving u'=Cu+G and x''=Ax+F.

Methods to solve homogeneous dynamical systems u'=Cu x'=x-5y, y'=x-y, x(0)=a, y(0)=b. Cayley-Hamilton method. Laplace resolvent. Eigenanalysis method. Exponential matrix using maple Putzer's method Spectral methods [ch8] Methods to solve for a particular sol u'=Cu+G x'=x-5y+5, y'=x-y +10, x(0)=a, y(0)=b. Laplace resolvent. Undetermined coefficients Equilibrium solution method Variation of parameters Computer algebra system methods Direct methods to solve u''=Au x''=x-5y, y''=x-y, x(0)=x'(0)=0, y(0)=y'(0)=1. Laplace resolvent. Cayley-Hamilton method. Eigenanalysis method.

Lecture: spectral theory, Putzer's formula and related topics ch8

Fundamental matrix. How to find the exponential matrix.

Systems u'=Au+F. Undetermined coefficients. Variation of parameters.

Variation of parameters example.

Slides: Railway cars, earthquakes.

Mechanical Resonance. Pure and practical resonance. mx''+cx'+kx=F0 cos(omega t)

Electrical resonance LI''+RI'+I/C=F0 omega cos(omega t). [typo corrected 15apr2009]

The resonance formulas omega=sqrt(k/m-c^2/(2m^2)) and omega = 1/sqrt(LC)

Theorem. The homogeneous solution for ay''+by'+cy=0 with a,b,c>0 constants has limit zero at x=infinity. This implies that ay''+by'+cy=f(x) has a unique T-periodic solution, if f(x+T)=f(x) for all x [f is T-periodic].

Theorem. The solution for ay''+cy=0 with a,c>0 constants is harmonic, which implies a fixed amplitude periodic oscillation of frequency sqrt(c/a).

Theorem. The equation ay''+by'+cy=f(x), a,b,c>0 constants, f(x) a periodic linear combination of atoms, has a unique periodic solution y(x) of the same period as f(x). The solution y(x) can be found by undetermined coefficients or Laplace theory.

Resonance curve for practical mechanical resonance.

Electrical resonance. Impedance, reactance. Steady-state current amplitude. Transfer function. Input and output equation.

Reference: Edwards-Penney, Differential Equations and Boundary Value Problems, 4th edition, section 3.7 [math 2280 textbook]. Extra pages supplied by Pearson with bookstore copies of the 2250 textbook. Also available as a xerox copy in case your book came from elsewhere. Check-out the 2280 book in the math library.

Lecture: Systems of two differential equations, Cramer's Rule, matrix inversion methods. The Laplace resolvent method for systems.

Example: Solving a 2x2 dynamical system using Laplace's resolvent method.

Study of u'=Au, u(0)=vector([2,1]), A=matrix([[2,3],[0,4]]).

Lecture: Cayley-Hamilton Theorem, Cayley-Hamilton Method to solve u'=Au.

Proof of Ziebur's lemma: the components of u in u'=Au are linear combinations of the atoms created by Euler's theorem applied to the roots of the characterisitic equation det(A-rI)=0.

Survey of Methods for solving a 2x2 dynamical system:

Cayley-Hamilton method for u'=Au Def: u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n) Def: Atoms constructed by Euler's theorem from roots of det(A-rI)=0 Def: vectors vec(c_1), ..., vec(c_n) are not arbitary. They are determined from A and u(0). Algorithms Wed-Fri. Laplace resolvent L(u)=(s I - A)^(-1) u(0) Eigenanalysis u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2 Putzer's method for the 2x2 matrix exponential. Solution of u'=Au is: u(t) = exp(A t)u(0) Def: exp(A t) = r1(t) I + r2(t) (A-lambda_1 I), Def: lambda_1 and lambda_2 are the roots of det(A-lambda I)=0. Def: r1'=lambda_1 r1, r1(0)=0; r2'= lambda_2 r2 + r1, r2(0)=0

Cayley-Hamilton method for systems u'=Au. (pdf slides)

Laplace method for systems, resolvent. (pdf slides)

Methods for second order systems, laplace resolvent, eigenanalysis, Cayley-Hamilton. (pdf slides)

Lecture: Topics from Laplace theory, linear systems and electric circuits.

Brine tank models. Recirculating brine tanks. Pond pollution. All are 3x3 system applications that can be solved by Laplace methods.

Brine tank cascade. Brine tank recyling. (pdf slides)

Lecture: Fourier's Model. Intro to eigenanalysis, ch6. Examples and motivation. Fourier's model. History. J.B.Fourier's 1822 treatise on the theory of heat. The rod example.

Physical Rod: a welding rod of unit length, insulated on the lateral surface and ice packed on the ends. Define f(x)=thermometer reading at loc=x along the rod at t=0. Define u(x,t)=thermometer reading at loc=x and time=t>0. Problem: Find u(x,t). Fourier's solution: f(x) = 17 sin (pi x) + 29 sin(5 pi x) = 17 v1 + 29 v2 Packages v1, v2 are vectors in a vector space V of functions on [0,1]. Fourier computes u(x,t) by re-scaling v1, v2 with numbers Lambda_1, Lambda_2 that depend on t. This idea is called Fourier's Model. u(x,t) = 17 ( exp(-pi^2 t) sin(pi x)) + 29 ( exp(-25 pi^2 t) sin (5 pi x)) = 17 (Lambda_1 v1) + 29 (Lambda_2 v2)

References on Fourier's model, eigenanalysis.

What's eigenanalysis slides 2008 (pdf)

Eigenanalysis-I manuscript S2009 (pdf)

Eigenanalysis-II manuscript S2009. Telecom example. stochastic matrices. (pdf)

Lecture: Algebraic Eigenanalysis, ch6. Calculation of eigenpairs to produce Fourier's model.

Connection between Fourier's model and a diagonalizable matrix.

How to find the variables lambda and v in Fourier's model using determinants and frame sequences.

algebraic eigenanalysis slides 2008 ( pdf)

Eigenanalysis-I manuscript S2009 (typeset 20 pages, 200k pdf)

Solved in class: examples similar to the problems in 6.1 and 6.2.

Slides and problem notes exist for 6.1 and 6.2 problems. See the web site.

References on Fourier's model, eigenanalysis.

What's eigenanalysis slides 2008 ( pdf)

algebraic eigenanalysis slides 2008 ( pdf)

Eigenanalysis-2 slides f2008 (pdf)

Eigenanalysis-I manuscript S2009 (typeset 20 pages, 200k pdf)

Cayley-Hamilton topics. Computing powers of matrices. Stochastic matrices.

Example of 1984 telecom companies MCI, SPRINT, ATT with discrete dynamical system u(n+1)=A u(n) with matrix A stochastic.

History of telecom companies (text)

Google. Lawrence Page's pagerank algorithm, google web page rankings, relation to the power method, stochastic matrices and eigenanalysis. Reference:

Where to find articles about Page's algorithm. (txt)

Projection: glass-breaking video.

More on resonance, including practical resonance theory.

Wine glass breakage (QuickTime MOV)

Wine glass experiment (12mb mpg 2min video)

Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg 4min video)

Maple lab on the Tacoma narrows (extra credit).

Lecture slides on the Cayley-Hamilton Method for solving u'=Au.

Cayley-Hamilton method for systems u'=Au. (pdf slides)

Lecture: EPbvp7.6, Delta function, Heaviside function. From EPbvp 7.6 supplement. See also web manuscript on this topic.

Lecture: Variation of parameters.

References: 5.5 variation of parameters formula (33).

Second order variation of parameters slides 2008 (pdf)

Second order variation of parameters (typeset, 6 pages pdf)

Start section 5.5 undetermined coefficients today. You will learn the basic algorithm to find y_p(x) using the method of undetermined coefficients. The method applies to constant-coefficient De with forcing term f(x)=linear combination of atoms.

Such DE can also be solved by Laplace theory. Even the calculation of y_p can be aided by using Laplace theory to find the corrected trial solution.

References: Sections 5.4, 5.6. Forced oscillations.

Forced vibrations, undamped case, slides 2008 (pdf)

Forced vibrations, damped case, slides 2008 (pdf)

Forced vibrations and resonance, slides 2008 (pdf)

Undetermined coefficients example, cafe door, pet door, phase-amplitude, resonance slides F2007

References: Higher order DE

Picard's Theorem for systems, slides 2008 (pdf)

How to solve linear DE, examples of orders 1 to 4, Euler's theorem explained. slides F2008 (pdf)

How to solve linear DE, slides S2008 (pdf)

Solving linear DE, examples of orders 1 to 4, slides 2008 (pdf)

Basic Reference: First order constant coefficient recipe, structure of solutions, superposition (slides, 3 pages pdf)

Basic Reference: First order constant coefficient recipe + theory + variation of parameters and undetermined coefficients (typeset, 11 pages pdf)

Second order constant coefficient recipe + theory (typeset, 7 pages pdf)

Problem session on 10.1, 10.2, 10.3, 10.4, 10.5, EPbvp7.6.

Get EPbvp7.6 from the math library: check out the 2280 book and hit the xerox at Kinkos. This is the packet bundled with E&P 3rd edition.

Solved problems 10.2-16, 10.5-4,22,28. EPbvp7.6-8.

The second shifting theorem, applications. The periodic function theorem, application. Ramp and pulse calculations. Heaviside and the unit step.

Answers to partial fraction questions. Resolvent method. How to model impulses.

Discussion of Riemann-Stieltjes integration theory and the delta function.

The meaning of delta in a model. Modeling hammer hits. How to use Laplace tables to avoid advanced calculus RS-integration. Delta function and the second shifting theorem.

Silde:

Piecewise-defined functions and Laplace theory slides F2008 (pdf)

Lecture: undetermined coefficients.

How to find the homogeneous solution yh from a characteristic equation.

Finding the redundant trial solution from g(x) = x^n f(x).

Given a trial solution with undetermined coefficients, find a system of equations for d1, d2, ... and solve it.

Reporting y_p from the trial solution and the answers d1, d2, d3, ...

Finding the non-redundant trial solution from g(x) = f(x) and the

Finding a trial solution with fewest symbols.

Relation between the non-redundant trial solution and the book's table that uses the mystery factor x^s.

Sita runs the Thursday lab session. Maple 5 discussed. Exam 2 problem 5 at the end of the hour. You may attend either 7:30 or 10:45 to take the exam.

Lecture: 5.6. Equations mx''+kx=F0 cos(omega t) for pure resonance and beats, equation mx''+cx'+kx=F0 cos(omega t) for practical resonance.

Formulas for the unique periodic solution

x=A cos(omega t) + B sin(omega t) and its amplitude

C = F0/sqrt(delta), where delta=(k-m omega)^2+(c omega)^2

Pure and practical resonance formula

omega = sqrt(k/m - c^2/(2m^2))

THEOREM. The homgeneous solution has limit zero at infinity for c>0.

THEOREM. There is a unique periodic solution obtained by undetermined coefficients when c>0 and the period is 2 PI / omega. This solution is the observable or steady state. All solutions of the homogeneous equation are transient.

In the Laplace solution, the output L(x) equals the transfer function time the Laplace of the input F0 cos(omega t). It is possible to find the steady state solution by dropping all negative exponential terms in the Laplace solution.

Drill: Graphics for beats [x=sin(10 t)sin(t/2)], slowly-oscillating envelope, rapidly oscillating harmonic with time-varying amplitude.

Lecture: Pure resonance and explosion of the solution, versus practical resonance, where omega is tuned to make C(omega)=maximum among all input frequencies.

Resonance examples: Soldiers marching in cadence, Tacoma narrows bridge, Wine Glass Experiment. Theodore Von Karman and vortex shedding. Cable model of the year 2000. Resonance explanations.

Plot of C(omega) versus omega. Practical resonance as the high point on a graphic.

Reading: Continue into chapter 10. We return to 5.6, ch6, ch7, ch8, ch9 later.

Lecture: Introduction to Laplace's method. The method of quadrature for higher order equations and systems. Calculus for chapter one quadrature versus the Laplace calculus. The Laplace integrator dx=exp(-st)dt. The Laplace integral abbreviation L(f(t)) for the Laplace integral of f(t). Lerch's cancellation law and the fundamental theorem of calculus.

Def: Direct Laplace transform == Laplace integral == int(f(t)exp(-st),t=0..infinity) == L(f(t)).

Linearity. The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).

Laplace theory references

Intro to Laplace theory. L-notation. Forward and backward table. Examples. (pdf slides)

Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems. (pdf slides)

Laplace theory typeset manuscript 2008 (49 pages pdf)

Heaviside's method 2008, typeset (pdf)

Ch10 Laplace solutions [scanned], 10.1 to 10.4 (9 pages, 2mb, pdf)

Lecture: Forward table. Backward table. Extensions of the Table.

Basic Laplace theory. Shift theorem. s-diff theorem. Parts theorem.

Lecture: Solving differential equations by Laplace's method.

Laplace rules and the brief table [Laplace calculus].

Lecture: Slides on Laplace theory. Worked examples and problems 10.1-18, 10.1-28. Using trig identities [sin 2u = 2 sin u cos u, etc].

Intro to Laplace theory. L-notation. Forward and backward table. Examples. (pdf slides)

Reference: Heaviside's method 2008, typeset

Lecture: Solving initial value problems by LaPlace's method. Details of the backward table and the forward table. Information about the equivalence of the inverse of L and Lerch's theorem.

Example: y'=-1, y(0)=0.

Example: y'=-1, y(0)=1.

Example: y''=-1, y(0)=y'(0)=0.

Example: y''+y'=-1, y(0)=y'(0)=0.

Use of inverse Laplace and Lerch's theorem.

Partial fraction expansions suited for LaPlace theory.

History of Laplace calculus and Newton Calculus.

Newton integral calculus and Laplace calculus. Laplace method. (pdf slides)

Laplace theory references

Intro to Laplace theory. L-notation. Forward and backward table. Examples. (pdf slides)

Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems. (pdf slides)

Laplace rules. Includes: Lerch, Linearity, t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution. (pdf slides)

Reference: Laplace theory typeset manuscript 2008 (49 pages pdf)

Reference: Heaviside's method 2008, typeset (pdf)

Ch10 Laplace exercise solutions [scanned], 10.1 to 10.4 (9 pages, 2mb, pdf)

Projection Slides:

Newton integral calculus and Laplace calculus. Laplace method. (pdf slides)

Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems. (pdf slides)

Laplace rules. Includes: Lerch, Linearity, t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution. (pdf slides)

Laplace method for systems, resolvent. (pdf slides)

Brine tank cascade. Brine tank recyling, home heating. (pdf slides)

Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation. (pdf slides)

Lecture: Review and drill on the forward and backward table, Laplace rules.

Examples of forward and backward table calculations. Harmonic oscillator.

Design of physical models from Laplace models instead of differential equations.

The transfer function. Input and output in the s-domain.

Periodic function theorem. Convolution theorm.

Example: The algebraic solution of a DE in integral form from the convolution theorem:

x''+x=F(t),x(0)=x'(0)=0 ==> x(t) = int(sin(t-u)F(u)du,u=0..t).

Details of proof for the convolution theorem.

Periodic waves used in engineering. Step, Ramp, Rectified sine wave, sawtooth, staircase.

Drill problems: More on ramp, sawtooth, staircase, rectified sine.

Exam 2. Covers chapters 3, 4 and 5.1, 5.2, 5.3.

Lecture: Partial fractions, Heaviside method, shortcuts, failsafe method for partial fractions [==sampling method], the Method of Atoms, Heaviside coverup.

Review and Drill: How to write up a solution which postpones partial fraction evaluation of constants to the end.

Lecture: Partial fraction methods for complex roots and roots of multiplicity higher than one.

Partial Fractions Examples: How to deal with complex factors like s^2+4. Heaviside's coverup method and how it works in the case of complex roots.

Problem: Write a 2x2 dynamical system as a vector-matrix equation u'=Au.

Problem: Solve a 2x2 dynamical system in vector-matrix form u'=Au.

Review: Expansions of periodic functions as s-domain Laplace transforms. The periodic function theorem.

Laplace of the square wave, tanh function.

Home heating example.

Home heating: attic, main floor, basement. (pdf slides)

Laplace resolvent method for systems of equations.

After spring break:

Rotating machine with one piston and massive flywheel - damping design.

Dimension. Basis for linear system Ax=0 from the last frame algorithm. Partial derivatives and bases.

Proofs involving subspaces for vector spaces V whose data item packages are functions or abstract vectors.

Theorem: pivot columns are independent and non-pivot columns are linear combinations of the pivot columns.

Theorem: rank(A)=rank(A^T).

Theorem: A set of nonzero pairwise orthogonal vectors is linearly independent.

In-class solutions. Please read the problem notes, ch4.

Problem session 4.3, 4.4, 4.7.

Solutions to 4.3-18,24, 4.4-6,24 and 4.7-10,22,26.

Web References:

Lecture slides on Vector spaces, Independence tests. (pdf)

Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)

Lecture slides on orthogonality, independence of orthogonal sets, Cauchy-Schwartz, Pythagorean Identity (pdf)

How to solve linear DE, examples of orders 1 to 4, Euler's theorem explained. slides 2008 (pdf)

Higher order constant equations, homogeneous and non-homogeneous structure. Superposition. Picard's Theorem. Solution space structure. Dimension of the solution set.

Euler's theorem. What to do with complex roots and conjugate pairs of factors (r-a-ib), (r-a+ib).

How to solve examples like y''=0, y''+3y'+2y=0, y''+y'=0, y'''+y'=0. The formula exp(i theta)=cos(theta) + i sin(theta). How to solve homogeneous equations: use Euler's theorem to find a list of n distinct atoms that are solutions to the equation. Specific examples for first, second and higher order equations.

How to use Euler's theorem to construct atoms of a linear differential equation.

Common errors in solving higher order equations.

Solved examples like the 5.1,5.2,5.3 problems.

Identifying atoms in linear combinations.

Lecture: Second order equation. Homogeneous equation.

Harmonic oscillator example y'' + y=0.

Picard's theorem. Dimension of the solution space. Structure of solutions.

Non-homogeneous equation. Forcing term. Nth order equations.

Solution space theorem for linear differential equations.

Superposition. Independence and Wronskians.

Definition of atom. Independence of atoms.

Main theorem on constant-coefficient equations [Solutions are linear combinations of atoms].

Euler's substitution y=exp(rx). Shortcut to finding the characteristic equation.

Euler's basic theorem: y=exp(rx) is a solution <==> r is a root of the characteristic equation.

Euler's multiplicity theorem: y=x^n exp(rx) is a solution <==> r is a root of multiplicity n+1 of the characteritic equation.

How to solve any constant-coefficient homogeneous differential equation.

Picard's Theorem for higher order DE and systems.

Picard's Theorem for systems, slides 2008 (pdf)

How to solve linear DE, examples of orders 1 to 4, Euler's theorem explained. slides 2008 (pdf)

Reference: How to solve linear DE, slides 2008 (pdf)

Reference: Solving linear DE, examples of orders 1 to 4, slides 2008 (pdf)

Basic Reference: First order constant coefficient recipe, structure of solutions, superposition (slides, 3 pages pdf)

Basic Reference: First order constant coefficient recipe + theory + variation of parameters and undetermined coefficients (typeset, 11 pages pdf)

Ch5. Constant coefficient recipe (typeset, 2 pages, pdf)

Second order constant coefficient recipe + theory (typeset, 7 pages pdf)

Undetermined coefficients, cafe door, pet door, phase-amplitude, resonance slides F2007

Lecture: Constant coefficient equations with complex roots. How to solve for atoms when the characteristic equation has multiple roots or complex roots.

Applying Euler's theorems to solve a DE.

Examples of order 2,3,4. Exercises 5.1, 5.2, 5.3.

Applications. Spring-mass system, harmonic oscillation, damped and undamped systems, forced systems. RLC circuit equation.

Spring-mass DE and RLC-circuit DE derivations. Electrical-mechanical analogy.

Drill: Sampling in partial fractions.

Drill: Method of atoms in partial fractions.

Drill: Heaviside's coverup method.

Lecture: Second order and higher order differential Equations.

Picard theorem for second order equations, superposition, solution space structure, dimension of the solution set.

Euler's theorem. Quadratic equations again. Constant-coefficient second order homogeneous differential equations. Characteristic equation and its factors determine the atoms for a second order linear DE.

Sample equations: y''=0, y''+2y'+y=0, y''-4y'+4y=0, y''+y=0, y''+3y'+2y=0, mx''+cx'+kx=0, LQ''+RQ'+Q/C=0.

How to solve for c1, c2, etc when initial conditions are given.

Theory of equations and 5.3-32.

Solving a DE when the characteristic equation has complex roots.

Higher order equations or order 3 and 4.

Damped and undamped equations. Phase-amplitude form.

Partly solved 5.4-34. The DE is 3.125 x'' + cx' + kx=0. The characteristic equation is 3.125r^2 + cr + kr=0 which factors into 3.125(r-a-ib)(r-a+ib)=0 having complex roots a+ib, a-ib. Problems 32, 33 find the numbers a, b from the given information. This is an inverse problem, one in which experimental data is used to discover the differential equation model. The book uses its own notation for the symbols a,b: a ==> -p and b ==> omega. Because the two roots a+ib, a-ib determine the quadratic equation, then c and k are known in terms of symbols a,b.

Finding 2 atoms from one complex root. Why the complex conjugate root identifies the same two atoms.

Equations with both real roots and complex roots.

An equation with 4 complex roots. How to find the 4 atoms.

Review and Drill.

The RLC circuit equation and its physical parameters.

Spring-mass equation mx''+cx'+kx=0 and its physical parameters.

Solving more complicated homogeneous equations. Example: Linear DE given by roots of the characteristic equation. Example: Linear DE given by factors of the characteristic polynomial. Example: Construct a linear DE of order 2 from a list of two atoms that must be solutions. Example: Construct a linear DE from roots of the characteristic equation. Example: Construct a linear DE from its general solution.

Lecture: 5.4 Damped and undamped motion. Pendulum, harmonic oscillations, spring-mass equation, phase-amplitude conversions from the trig course. Beats. Pure resonance. Cafe door. Pet door.

overdamped, critically damped and under damped behavior, pseudoperiod.

Problems solved in class: all of 5.2, 5.3 and 5.4-20,34

References: Sections 5.4, 5.6. Forced oscillations.

Forced vibrations, undamped case, slides 2008 (pdf)

Forced vibrations, damped case, slides 2008 (pdf)

Forced vibrations and resonance, slides 2008 (pdf)

Undetermined coefficients example, cafe door, pet door, phase-amplitude, resonance slides F2007

Lecture: Introduction to Laplace theory. Newton and Laplace portraits.

Method of quadrature. Comparison of Newton calculus and Laplace calculus.

Direct Laplace transform == Laplace integral.

Laplace theory references week 10-11.

Intro to Laplace theory. L-notation. Forward and backward table. Examples. (pdf slides)

Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems. (pdf slides)

Laplace theory typeset manuscript 2008 (49 pages pdf)

Heaviside's method 2008, typeset (pdf)

Ch10 Laplace solutions [scanned], 10.1 to 10.4 (9 pages, 2mb, pdf)

Didn't do 3.6-60? Do extra credit from ch3.

Abstract vector spaces. Working set == subspace. Data set == Vector space.

Lecture: Chapter 4, sections 4.1 and 4.2.

Web references for chapter 4.

Lecture slides on Vector spaces, Independence tests. (pdf)

Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)

Vector space, Independence, Basis, Dimension, Rank (typeset pdf)

Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (4 pages, 480k pdf)

Vectors==packages of data items. Vectors are not arrows.

Examples of vectors: digital photos, Fourier coefficients, Taylor coefficients, sols to DE like y=2exp(-x^2) [DE is y'=-2xy, y(0)=2].

Data recorder example.

More on vector spaces and subspaces: detection of subspaces and data sets that are not subspaces.

Theorems: Subspace criterion, Kernel theorem, Not a subspace theorem.

Use of theorems 1,2 in section 4.2. Solutions of problems 4.1, 4.2.

Example in class: Subspace Shortcut. Avoid using the subspace criterion on the set S in R^3 defined by x+y+z=0, by writing it as Ax=0, followed by applying the kernel theorem (thm 2 page 239 or 243 section 4.2 of Edwards-Penney).

Subspaces. Using the kernel theorem. Problems 4.1,4.2.

More on the toolkit. Vectors as packages of data items. Examples of vector packaging in applications. The kernel: sols of Ax=0.

Vector spaces. The toolkit of 8 properties. Subspaces. Subspace criterion (Thm 1, 4.2). Kernel theorem (Thm 2, 4.2).

Independence and dependence. Example c1 + c2 x = 1 + 2x ==> c1=1, c2=2.

Atoms and solutions of differential equations. Geometric test for one or two vector independence.

Lecture: More on independence. Algebraic tests. Geometric tests.

Independence tesats: Rank test. Determinant test. Pivot theorem. Sampling test. Wronskian test. Pivot Theorem.

General solutions with a minimal number of terms.

Data recorder example and data conversion to fit physical models.

Basis == independence + span. Independence of atoms.

Function spaces. General solution and shortest answer.

Web References:

Lecture slides on Vector spaces, Independence tests. (pdf)

Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)

Lecture: Section 4.4 and 4.7. Some part of 4.5, on the pivot theorem.

How to develop notation for functions that parallels column vectors.

Hybrid vector spaces and RLC circuits.

Pivot theorem and other independence tests. Proof of the pivot theorem.

Problem session on ch4 problems. Especially 4.3 problems: Illustration of how to do abstract independence arguments using vector packages, without looking inside the packages.

Applications of the rank test and determinant test.

Applications of the pivot theorem to find a largest set of independent vectors.

Bases. Equivalence of bases. A test for equivalent bases. Maximum set of independent vectors from a list.

Sita and Loc, session on exam 2 review, maple lab 2, ch4 problems.

Lecture on 4.3, 4.4, 4.7 problems, including complete solutions. Examples of how to use the subspace criterion to prove a set S in V is a subspace.

Last frame algorithm and the vector general solution. Basis of solutions to a homogeneous system of linear algebraic equations. Bases and partial derivatives on the invented symbols t1, t2, ...

The equation y'' 10y'=0. How to solve it based upon chapter 1 and the ideas in midterm problem 1(d). Idea: let v=x'(t) to get a frst order DE in v and a quadrature equation x'(t)=v(t).

Atoms. Base atoms are 1, exp(a x), cos(b x), sin(b x), exp(ax)cos(bx), exp(ax)sin(bx). Define: atom=x^n(base atom).

Theorem. Atoms are independent.

Theorem. Solutions of constant-coefficient homogeneous differential equations are linear combinations of atoms.

Picard's theorem says that nth order equations have a solution space of dimension n.

Euler's theorem: y=exp(rx) is a solution of ay''+by'+cy=0 <==> r is a root of the characteristic equation ar^2+br+c=0.

Shortcut: The characteristic equation can be synthetically formed from the differential equation ay''+by'+cy=0 by replacing y ==> 1, y' ==> r, y'' ==> r^2.

The equation y''+10y'=0 has characteristic equation r^2+10r=0 with roots r=0, r=-10. Then Euler's theorem says exp(0x) and exp(-10x) are solutions. By the dimension theory, 1, exp(-10x) are a basis for the solution space of the differnetial equation and then the general soution is y = c1 (1) + c2 (exp(-10x)).

Due today, 3.4-20,30,34,40. See problem notes on the web.

Elementary matrices. Inverses of elementary matrices.

Solving B=E3 E2 E1 A for matrix A = (E3 E2 E1)^(-1) B.

How to do 35.-16 in maple. Maple ans checks.

> with(linalg):#3.5-16 > A:=matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]); > A1:=augment(A,diag(1,1,1)); > rref(A1); > B:=inverse(A); > A2:=addrow(A1,1,2,1); > A3:=addrow(A2,1,3,-2); > evalm(A&*B);

Fundamental theorem on frame sequences

THEOREM. If A2 is the frame just after frame 1, A1, then A2=E A1 where E is the elementary matrix built from the identity matrix I by applying one toolkit operation combo(s,t,c), swap(s,t) or mult(t,m).

THEOREM. If A is the first frame and B a later frame in a sequence, then there are elementary swap, combo and mult matrices E_1 to E_n such that the frame sequence A --> B can be writen as the matrix multiply equation

B=E_n E_{n-1} ... E_1 A.

THEOREM. The 4 rules for computing any determinant can be compressed into two rules, (1) The triangular rule and (2) det(EA)=det(A)det(A) where E is an elementary combo, swap or mult matrix.

Lecture: Introduction to 3.6 determinant theory and Cramer's rule.

Sarrus' rule for 2x2 and 3x3. General Sarrus' rule with n-factorial arrows.

Lecture: Adjugate formula for the inverse. Review of Sarrus' Rules.

Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)

Determinants and Cramer's Rule (manuscript, 11 pages, 140k pdf)

Determinant slides 2008 (pdf)

Lecture: Methods for computing a determinant. Sarrus' rule, four rules for determinants, cofactor expansion, hybrid methods.

In-class examples.

Triangular rule [one-arrow Sarrus' rule], combo, swap and mult rules.

College algebra determinant definition and Sarrus' rule for 2x2 and 3x3 matrices.

Examples where det(A) can be comupted.

Determinant values for elementary matrices: det(E)=1 for combo(s,t,c), det(E)=m for mult(t,m), det(E)=-1 for swap(s,t).

Main theorems:

- Computation by the 4 rules, cofactor expansion, hybrid methods.
- Determinant product theorem det(AB)=det(A)det(B).
- Cramer's Rule for solving Ax=b.
- Adjugate inverse formula inverse(A) = adjugate(A)/det(A).

Lecture: Cofactor expansion. Hybrid methods. Frame sequences and determinants. Formula for det(A) in terms of swap and mult operations.

Hybrid methods to compute a determinant. In-class example. Special theorems for determinants having a zero row, duplicates rows or proportional rows.

How to use the 4 rules to compute det(A) for any size matrix. Computing determinants of sizes 3x3, 4x4, 5x5 and higher.

Elementary matrices and determinants. Determinant product rule.

Cramer's rule.

How to form minors, cofactors and the adjugate matrix.

Statements of the determinant product theorem and the adjugate inverse theorem.

How to reduce the Four rules [triang, swap , combo, mult] to Two Rules using the determinant product theorem det(AB)=det(A)det(B).

Digitial photos and matrices.

Web Reference: Image sensors, digital photos, checkerboard analogy, visualization of matrix addition and scalar multiplication.

Digital photos and matrix operations slides S2009

Digital photo slides F2008

Lecture: Cofactor rule and the adjugate matrix.

How to compute a cofactor expansion.

main theorems.

A adj(A) = adj(A) A = det(A) I

Cramer's rule for Ax = b.

x_1 = delta_1/delta, ... , x_n = delta_n/delta

Adjugate matrix and inverse formulas.

Thursday Lecture [Loc and sita]:

Maple Lab 3 Spring 2009 (pdf)

Maple L3 snips Spring 2009 (maple text)

Maple Worksheet files: In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...". Some browsers require SHIFT and then mouse-click. Open the saved file in xmaple.

Maple L3 snips worksheet Spring 2009 (maple .mws)

Numerical DE coding hints (txt)

The actual symbolic solution derivation and answer check are submitted as L3.1. Confused? Follow the details in the next link.

Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)

Do not submit any work for L3.1 with decimals! Only 1.4 methods are to appear.

The numerical work, Euler, Heun, RK4 parts are L3.2, L3.3, L3.4.

Confused about what to put in your L3.2 report? Do the same as what appears in the sample report for 2.4-3 (below).

Sample Report for 2.4-3. Includes symbolic solution report. (pdf 3 pages, 350k)

Sample maple code for Euler, Heun, RK4 (maple .mws)

Sample maple code for exact/error reporting (maple .mws)

Additional reference, probably not needed:

Report details on 2.4,2.5,2.6 prob 6 (pdf)

Cofactor rules imbedded in the formula det(A)I = A adj(A) = adj(A) A.

Determinant slides 2008 (pdf)

Examples in class: (1) solve a 2x2 system by Cramers rule. (2) Find entry in row 3, col 2 of the inverse of A = adj(A)/det(A) as a quotient of 2 determinants. (3) Find det(A) from A and adj(A).

Problem session on 3.6 exercises.

Review: 2x2, 3x3, 4x4 determinant evaluation examples. Cramers rule example. Adjugate example. Computing entry 2,3 in adj(A) or inverse(A).

Lecture: Maple lab problem L2.1 discussed today. Solution projected for L2.1 and L2.4.

Drill: last frame algorithm and the scalar and vector forms of the solution. Equality of vectors as the method for converting scalar solutions to vector solutions, and conversely.

Lecture: General structure of linear systems. Superposition.

Properties of matrices. Matrix multiply rules. Matrix multiply Ax for x a vector. Linear systems as the matrix equation Ax=b.

Lecture: Fixed vectors, physics vectors i,j,k, engineering vectors (arrows), Gibbs vectors. Parallelogram law. Head-tail rule.

Lecture: The 8-property toolkit for vectors. Vector spaces. Read 4.1 in Edwards-Penney, especially the 8 properties.

Review of vector models is in the slide set

Slides on vector models and vector spaces 2008 (pdf)

Slides and examples for chapter 3

Linear equations, reduced echelon, three rules (typeset, 7 pages, pdf)

Three rules, frame sequence, maple syntax (typeset, 7 pages, pdf, 12 Oct 2006)

Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)

Elementary matrices, vector spaces, slides (8 pages, pdf, 12 Oct 2006)

Slides on matrix Operations (pdf)

Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)

Typeset manuscripts for ch3 and ch4

Linear equations, no matrices, DRAFT 2009 (typeset, 45 pages, pdf)

vectors and matrices (typeset, 14 pages)

Matrix equations (typeset, 12 pages)

Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)

Review and drill on linear equations: Unique solution case, infintely many solution case, last frame algorithm. Equality of vectors. Matrix-vector equations Ax=b. Vector form of a solution in both the unique solution case and the last frame algorithm case. Superposition. General solution X=X0+t1 X1 + t2 X2 + ... + tn Xn.

Review: The four vector models, vector spaces and the 8-property toolkit.

Lecture: Elementary matrices.

Theorem: If a frame sequence starts with A and ends with B, then B=(product of elementary matrices)A.

Web Reference: Elementary matrices

Elementary matrix slides S2009

Exam 1. First exam starts at 7am in JTB 140. Both 7:30 and 10:45 students are invited. Choose this time if you need extra time on exams. The second exam starts at 10:35am. Both 7:30 and 10:45 students are invited.

Discussion of 3.4 problems.

Due today, maple lab L2.1.

Lecture: How to write a frame sequence as a product of elementary matrices.

Lecture: How to compute the inverse matrix from inverse = adjugate/determinant and also by frame sequences. Inverse rules.

Web Reference: Construction of inverses. Theorems on inverses.

slides on rref inverse method S2009

Lecture: Ideas of rank, nullity, dimension in examples.

More on Rank, Nullity, dimension, 3 possibilities, elimination algorithm.

Slides on rank, nullity, elimination algorithm Feb 2009 (pdf)

Answer to the question: What did I just do, when I found rref(A)?

Due Monday: All of 3.4; see FAQ 3.4 for details.

Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with A in reduced echelon form. Apply the last frame agorithm then write the general solution in vector form.

Discussion of Cayley-Hamilton theorem [3.4-29] and how to solve problem 3.4-30.

Problem 3.4-29 is used in Problem 3.4-30. The result is the Cayley-Hamilton Theorem, a famous theorem of linear algebra which is the basis for solving systems of differential equations.

Problem 3.4-40 is the superposition principle for the matrix equation Ax=b. It is the analog of the differential equation relation y=y_h + y_p.

Lecture: 3.1, frame sequences, combo, swap, multiply, geometry

Due today: Due today ER-1, ER-2 [credit applied to maple L3.1, L4.1]

Prepare 3.1 problems for next collection.

References for chapter 3

Slides on Linear equations, reduced echelon, three rules (pdf)

Slides on Linear equations, unique solution case (pdf)

Slides on Linear equations, no solution case, signal equations (pdf)

Slides on Linear equations, infinitely many solution case, last frame algorithm (pdf)

Three rules, frame sequence, maple syntax (typeset, 7 pages, pdf, 12 Oct 2006)

Frame sequence and general solution, 3x3 system (1 page, pdf, 28-Sep-2006)

Linear algebra, no matrices, DRAFT Feb 2009 (typeset, 44 pages, pdf)

Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)

vectors and matrices (typeset, 14 pages)

Matrix equations (typeset, 12 pages)

Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)

Elementary matrices, vector spaces, slides (8 pages, pdf, 12 Oct 2006)

Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)

Lecture: 3.1, 3.2, 3.3, frame sequences, general solution, three possibilities.

A detailed account of the three possibilities. How to solve a linear system using the tookit [swap, combo, mult] and frame sequences, for the unique solution case, no solution case and infinitely many solution case. Examples.

Slides for this lecture.

Slides on Linear equations, unique solution case (pdf)

Slides on Linear equations, no solution case, signal equations (pdf)

Slides on Linear equations, infinitely many solution case, last frame algorithm (pdf)

Sample solution ER-1 [same as L3.1] (jpg)

In 3.2 solutions, back-substitution should be presented as combo operations in a frame sequence, not as isolated algebraic jibberish.

Computer algebra systems and error-free frame sequences.

How to program maple to make a frame sequence without errors.

Problem 3.2-24: See Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)

Beamer Slides on three possibilities with symbol k, Sept2007 (9 pages,pdf)

See also Example 10 in Linear equations, no matrices, DRAFT Feb2009 (typeset, 45 pages, pdf)

Prepare 3.3 problems 8, 18 for next time. Please use frame sequences to display the solution, as in today's lecture examples. It will be a sequence of augmented matrices. Yes, you may use maple to make the frame sequence and to do the answer check [rref(A);].

Answer checks should also use the online FAQ.

Lecture: 3.3 and 3.4.

Translation of equation models to (augmented) matrix models and back. Combo, swap and multiply for matrix models. Frame sequences for matrix models.

Review of rref or last frame test.

Review: Last frame algorithm. Drill: last frame algorithm and the scalar form of the solution.

Equation ideas can be used on a matrix A. View matrix A as the set of coefficients of a homogeneous linear system Ax=0. The augmented matrix B for this homogeneous system would be the given matrix with a column of zeros appended: B=aug(A,0).

Answer checks with matlab, maple and mathematica. Pitfalls.

Review of the three possibilities and frame sequence analysis to find the geberal solution.

Frame sequences with symbol k.

Slides on the 3 possibilities, rank, sytems with symbol k 11Feb2008 (pdf)

Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)

Matrix multiply, college algebra, examples.

Slides on matrix Operations (pdf)

Examples: how to multiply matrices on paper.

If you are unable to turn in this lab, then see the Ch2 Extra Credit problems, which contains 2 problems like L1.1 and L1.2.

Review and reading: Please study the slides on partial fractions.

Partial Fraction Theory 2008(125k pdf)

Nonlinear air resistance models F=kx'|x'|. Lecture slides on the reading assignment for 2.3, and the work of Isaac Newton on ascent and descent models for kinematics with air resistance.

Newton models, projectile slides 2008 (pdf)

Problem notes for 2.3-10. 2.3-20 are available,

For more details on the 2.3 problems,

Review of 2.2: Verhulst models with harvesting term. Differences in phase diagrams of non-autonomous systems. What new things we see in computer graphics of such models.

Intro to the Jules Verne problem and its solution.

Slides on the Jules Verne problem [reading: 2.3].

Earth to the moon slides 2008 (pdf)

Problems discussed: 2.3-10 and 2.3-20.

Problem notes for 2.3-10. 2.3-20 including sample maple code:

Reading assignment: proofs of 2.3 theorems in the textbook and derivation of details for the rise and fall equations with air resistance.

Lectures begin for 2.4, 2.5, 2.6 topics on numerical solutions.

Numerical DE slides 2008 (14 slides pdf)

Due today, 2.3: 10, 20.

Continued: Introduction to numerical solutions of quadrature problems y'=F(x), y(x0)=y0.

The examples used in maple labs 3, 4 are y'=-2xy, y(0)=2, y=2exp(-x^2) and y'=(1/2)(y-1)^2, y(0)=2, y=(x-4)/(x-2). Web notes (item 2 in the references below) contain the examples y'=3x^2-1, y(0)=2, y=x^3-x+2 and y'=exp(x^2), y(0)=2 with solution y=int(F,0..x)+y0, F(x)=exp(x^2).

Intro to Rect, Trap, Simp rules from calculus and Euler, Heun, RK4 rules from this course.

- References for numerical methods:
- Linear integrating factor method, Section 1.5, slides (pdf)
- Linear DE method, Section 1.5 slides: 1.5-3,5,11,33+Brine mixing (9 pages, pdf)
- Linear DE part I (Integrating Factor Method), (typeset, 8 pages, pdf)
- Linear DE part II (Variation of Parameters, Undetermined Coefficients), (typeset, 7 pages, pdf)
- How to do a maple answer check for y'=y+2x (TEXT 1k)
- Linear first order slides, integrating factor method (2 pages, pdf)

Due next week, Page 54, 1.5: 8, 18

### Week 1, Jan 12 to 16: Sections 1.1,1.2,1.3,1.4.

**12 Jan**: Details about exams and dailies.

Fundamental theorem of calculus. Method of quadrature [integration method in Edwards-Penney].

- Week 1 references (documents, slides)

Three Examples, Fundamental Theorem of Calculus, Method of quadrature, Decay law derivation, Background formulas. 6 slides, pdf.

Three Examples (pdf)

Three Examples, solved 1.2-1,2,5,8,10 by Tyson Black, Jennifer Lahti, GBG, 11 slides, pdf.

Log+exponential Background+Direction fields PDF Document (4 pages, 450k). Decay law derivation. Problem 1.2-2. Direction field examples.

Three Fundamental Examples introduced: growth-decay, Newton Cooling, Verhulst population. See Three Examples (pdf)

Background, 3 Examples, Decay Equation Derivation.

**13 Jan**: Collected exercise 1.2-2. Exercises 1.2-4, 1.2-6, 1.2-10 discussed in class. Slides projected: Tyson Black 1.2-1, Jennifer Lahti 1.2-2,10,

Sections 1.1, 1.2. Example for problem 1.2-1, similar to 1.2-2.

Started topic of partial fractions, to be applied again in 2.1-2.2. Heaviside's method. Fail-safe method.

Panels 1 and 2 in the answer check for an initial value problem like 1.2-2: y'=(x-2)^2, y(2)=1.

Answer checks. Proof that "0=1" and logic errors in presentations.

Maple tutorials start next week. Maple lab 1 is due soon, please print it from the link Maple Lab1.

More on the method of quadrature:

Drill problems and quadrature reference (pdf).

Integration details and how to document them using handwritten calculations like u-subst, parts, tabular. Maple and Matlab methods. Integral table methods.

**14 Jan**: Lecture on direction fields and existence-uniqueness 1.3. Discuss 1.3-8.

Collected in class Page 16, 1.2: 4, 6.

Integration theory examples. Method of quadrature. Parts, tables, maple.

Euler's directional field visualization, tools for using Euler's idea, reduction of an initial value problem to infinitely many graphics, showing the behavior of all solutions, without solving the differential equation.

Lecture on 1.2-8.

Direction field reference: Direction fields manuscript, 11 pages, pdf.

Threading edge-to-edge solutions is based upon two rules: (1) Solution curves don't cross, and (2) Threaded solution curves must match tangents with nearby arrows of the direction field. See the direction field document above for explanations.

For problem 1.3-8, xerox at 200 percent the textbook page and paste the figure. Draw threaded curves on this figure according to the rules in the direction field document above. Zoomed copy of Edwards-Penney exercise 1.3-8, to be used to produce HW copy.

**15 Jan**: Intro by Thursday TA staff. Discuss submitted work presentation ideas. Drill, examples, questions. Discuss problems 1.2-1.3. Discussion of Exam review plan for the semester.

**16 Jan**: Collected nothing.

Discussion of 1.3 problem. Distribution of maple lab 1 at web site.

How to thread curves on a direction field. Exercise 1.3-8. Zoomed copy of Edwards-Penney exercise 1.3-8, to be used to produce HW copy.

Picard-Lindelof Theorem, Peano Theorem, example y'=3(y-1)^(2/3), y(0)=1, like 1.3-14. The Picard-Lindelof theorem and the Peano theorem are found in this slide set: Peano and Picard Theory (3 pages, pdf).

For problem 1.3-14, see

Picard-Lindelof and Peano Existence theory manuscript, 9 pages, pdf

Peano and Picard Theory, 3 slides, pdf

Picard-Lindelof and Peano Existence Example, similar to 1.3-14, 1 slide, pdf

Summary of Peano, Picard and direction Fields [Jan 2008]

Peano, Picard, Direction Fields (slides, pdf)

For 1.3-14, a discussion of background material on**functions and continuity****Click here.**

For the write-up of 1.3-14 see part (a) of the link

Picard-Lindelof and Peano Existence Example (1 page, pdf).

See also examples in the summary of Peano, Picard and direction Fields [Jan 2008]

Peano, Picard, Direction Fields (slides, pdf)

Numerical DE slides 2008 (14 slides pdf)

Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)

Sample Report for 2.4-3 (pdf 3 pages, 350k)

ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)

The work for 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4. Details for lab 3:

Maple Lab 3 S2009 (pdf)

Maple L3 snips S2009 (maple text)

Maple Worksheet [.mws] files: In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...". Some browsers require SHIFT and then mouse-click. Open the saved file in xmaple or maple.

Maple L3 snips worksheet Spring 2009 (maple .mws)

Numerical DE coding hints (txt)

The actual symbolic solution derivation and answer check are in L3.1. Didn't do it earlier.? See similar details in the next link.

Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)

The numerical work using Euler, Heun, RK4 appears in L3.2, L3.3, L3.4.

Confused about what to put in your L3.2 report? Do the same as what appears in the sample report for 2.4-3 (below).

Sample Report for 2.4-3. Includes symbolic solution report. (pdf 3 pages, 350k)

Sample maple code for Euler, Heun, RK4 (maple .mws)

Sample maple code for exact/error reporting (maple .mws)

Discuss questions on maple Lab 2, especially parts 2,3,4.

How to present the solutions to the exam review problems ER-1 and ER-2 [identical to math-only problems L3.1, L4.1].

Discussed y'=3x^2-1, y(0)=2 with solution y=x^3-x+2. Dot tables, connect the dots graphic.

How to draw a graphic without knowing the solution equation for y. Main example y'=srqt(x)exp(x^2), y(0)=2. Making the dot table by approximation of the integral of F(x). Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.

Example for your study: The problem y'=x+1, y(0)=1 has a dot table with x=0, 0.25, 0.5, 0.75, 1 and y= 1, 1.25, 1.5625, 1.9375, 2.375. The exact solution y=1/2+(x+1)^2/2 has values y=1, 1.28125, 1.625, 2.03125, 2.5000. Try to determine how the dot table was constructed and identify which rule [Rect, Trap, Simp] was applied.

Symbolic solution exam review problems ER-1, ER-2 [duplicate maple L3.1, L4.1] are due next Monday.

Discussion of Euler, Heun, RK4 algorithms. Computer implementations.

Numerical work maple L3.2-L3.4, L4.2-L4.4 will be submitted after the spring break. All discussion of maple programs will be based in the Tuesday session. There will be one additional presentation of maple lab details in the main lecture.

Numerical DE slides 2008 (14 slides pdf)

Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)

Sample Report for 2.4-3 (pdf 3 pages, 350k)

ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)

Collected Page 41, 1.4: 18, 22, 26

Problem Notes on 1.5-34:

The expected model is x'=1/4-x/16, x(0)=20, using units of millions of cubic feet. The answer is x(t)=4+16 exp(-t/16).

Derivation of the model in 1.5-34 using x'=input rate - output rate.

Definition of concentration == amt/volume. Use of percentages in concentrations [0.25% concentration means 0.25/100 == concentration].

Superposition for y'+p(x)y=0. Superposition for y'+p(x)y=q(x).

Growth-Decay model y'=ky and its algebraic model y=y(0)exp(kx).

Pharmokinetics for drug transport [ibuprofen], brine tanks, pollution models. One-tank model. Two-tank and three-tank models. Recycled brine tanks and non-solvability by chapter 1 methods. Three lake pollution model [Erie, Huron, Ontario]. Linear cascades and their solution using the linear integrating factor method on successive equations.

Method (1) Linear integrating factor method. Method (2) Superposition and equilibrium solutions for constant-coefficient y'+py=q. Uses a shortcut for growth-Decay DE y'+py=0.

- References for linear DE:

Linear integrating factor method, Section 1.5, slides (pdf)

Linear DE method, Section 1.5 slides: 1.5-3,5,11,33+Brine mixing (9 pages, pdf)

Linear DE part I (Integrating Factor Method), (typeset, 8 pages, pdf)

Linear DE part II (Variation of Parameters, Undetermined Coefficients), (typeset, 7 pages, pdf)

How to do a maple answer check for y'=y+2x (TEXT 1k)

Linear first order slides, integrating factor method (2 pages, pdf)

Collected in class: Page 54, 1.5: 8, 18

Some more class discussion of 1.5-34.

Due Wed, Page 54, 1.5: 20, 34.

Drill on Section 1.5: Three linear examples: y'+(1/x)y=1, y'+y=x, y'+2y=1.

Intro to Maple lab 1: Theory of equations review, including quadratic equations, Factor and root theorem, division algorithm, recovery of the quadratic from its roots.

Due soon, Page 106, 2.3: 10, 20

Introduction to 2.1, 2.2 topics: autonomous DE, partial fraction methods. Discussion of 2.1-6,16.

Drill: classification separable, quadrature, linear.

Drill: Methods for solving first order equations: (1) Linear integrating (2) Superposition + equilibrium solution for constant-coefficient linear, (3) Quadrature method, (4) Variables Separable method, includes equilibrium solutions from G(y)=0 and non-equilibrium solutions from G(y) nonzero. factor method,

Reading on partial fractions [we study (1) sampling, (2) method of atoms, (3) Heaviside cover-up]:

Partial Fraction Theory 2008(125k pdf)

- References for 2.1, 2.2, 2.3:

Autonomous DE slides 2008 (pdf)

Newton models, projectile slides 2008 (pdf)

Earth to the moon slides 2008 (pdf)

Verhulst logistic equation (typeset, 5 pages, pdf)

Phase Line and Bifurcation Diagrams (includes "Stability, Funnel, Spout, and bifurcation") (typeset, 6 pages, 161k pdf)

ch2 sections 1,2,3 Slides: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (8 pages, pdf)

ch2 DEplot maple example 1 for exercises 2.2, 2.3 (1 page, 1k)

ch2 DEplot maple example 2 for exercises 2.2, 2.3 (1 page, 1k)

Lecture on 2.1, 2.2 problems. Discussion of 2.2-10,18.

Lecture on midterm 1 problem 5, in conjunction with 2.2-10,18. Maple L2.1 details. Drill on chapter 1 problems.

How to construct phase line diagrams. How to make phase plots. Discussion of 2.2-10,18.

Work on problems 2.1-6,16 and ask questions on Thursday.

Phase diagrams, calculus tools, DE tools, partial fraction methods. Next week: 2.3 and numerical DE topics 2.4 to 2.6.

Introduction to Newton models for falling bodies and projectiles. Newton's laws.

References for 2.1, 2.2, 2.3:

Autonomous DE slides 2008 (pdf)

Newton models, projectile slides 2008 (pdf)

Earth to the moon slides 2008 (pdf)

Verhulst logistic equation (typeset, 5 pages, pdf)

Phase Line and Bifurcation Diagrams (includes "Stability, Funnel, Spout, and bifurcation") (typeset, 6 pages, 161k pdf)

ch2 sections 1,2,3 Slides: 2.1-6, 2.1-16 (rabbit), 2.1-38, 2.2-4, 2.2-10, 2.3-9, 2.3-27+Escape velocity (8 pages, pdf)

ch2 DEplot maple example 1 for exercises 2.2, 2.3 (1 page, 1k)

ch2 DEplot maple example 2 for exercises 2.2, 2.3 (1 page, 1k)

Heaviside partial fraction method (4 pages, 86k)

Heaviside's method and Laplace theory (153k pdf)

Partial Fraction Theory 2008(125k pdf)

Content: Maple lab discussed: Lab 2 problem 1. Sources on www. Problems discussed from 2.1 to 2.3. Midterm sample is the F2008 exam, found on the web.

Links for maple lab 2:

maple Lab 2 F2008 (pdf)

maple worksheet text Lab 2 F2008

For more on superposition y=y_p_ + y_h, see Theorem 2 in the link Linear DE part I (8 pages pdf)

For more about home heating models, read the following link.

Linear equation applications, brine tanks, home heating (typeset, 12 pages, pdf)

Due today, Page 86, 2.1-6,16. Next time: 2.2-10,14

Lecture on stability theory.

Covered in class for 2.1, 2.2: theory of autonomous DE y'=f(y), stability of equilibrium solutions, funnel, spout, node, phase line diagram, phase diagram, unstable, classification of equil solutions.

Lecture: Introduction to Newton's force and friction models.

Free fall with no air resistance F=0.

Linear air resistance models F=kx'.

Non-linear air resistance models F=k|x'|^2.

The tennis ball problem. Does it take longer to rise or longer to fall?

Jules Verne problem. A rocket from the earth to the moon.

Drill: Direction fields, Two Rules, Picard and Peano Theorems.

Theory of separable equations, section 1.4.

Definition of separable DE. Some tests.

Examples: 1.4-6,12,18. See the Problem Notes for complete answers and methods.

Reference slides for separable DE.

Separable Equations 2008 slides, separability test, tests I and II (9 pages, pdf)

Separable Equations manuscript, classification (pdf)

1.4 Page 40 Exercise slides (4 pages, 500k)

How to do a maple answer check for y'=y+2x (TEXT 1k)

Drill: Quadrature, integral of du/(1+u^2), 2u du/(1+u^2). True and false trig formulas: arctan(tan(theta))=theta [false], tan(arctan(x))=x [true].

Solutions for 1.4-6,12,18. See also Problem Notes 1.4 at the web site.

Exercises Page 41, 1.4: 6, 12 are due next.

Theory of separable equations continued, section 1.4.

Separation test: F(x)=f(x,y0)/f(x0,y0), G(y)=f(x0,y), then FG=f if and only if y'=f(x,y) is separable. Basic theory discussed.

Basic theory: y(x) = H^(-1)( C1 + int(F)), H(u)=int(1/G,u0..u).

The solutions y=constant are called equilibrium solutions. Find them using G(c)=0. Non-equilibrium solutions from y'/G(y)=F(x) and a quadrature step. Implicit and explicit solutions. Discussion of answer checks for implicit solutions and also explicit solutions. Some troubles with explicit solutions of y'= 3 sqrt(xy) [1.4-6]. Separable DE with no equilibrium solutions. Separable DE with infinitely many equilibrium solutions. The list of answers to a separable DE. Influence of an initial condition to extract just one solution formula from the list.

Examples for Midterm 1 problem 2. y'=x+y, y'=x+y^2, y'=x^2+y^2

Example 1: Show that y'=x+y is not separable using TEST I or II (partial derivative tests).

Example 2: Find the factorization f=F(x)G(y) for y'=f(x,y), given

(1) f(x,y)=2xy+4y+3x+6 [ans: F=x+2, G=2y+3].

(2) f(x,y)=(1-x^2+y^2-x^2y^2)/x^2 [ans: F=(1-x^2)/x^2, G=1+y^2].

Collect in class Page 41, 1.4: 6, 12. Next time Page 41, 1.4: 18, 22, 26

Drill on the variables separable method. Discuss remaining 1.4 exercises.

Start 1.5, theory of linear DE y'=-P(x)y+Q(x). Integrating factor, the fraction that replaces two-termed expression y'+py.

Classification of y'=f(x,y): quadrature, separable, linear. Venn diagram of classes. Examples of various types.

Tests for quadrature (f_y=0) and linear (f_y indep of y) types.

Linear integrating factor method 1.5. Application to y'+2y=1 and y'+y=e^x. Testing linear DE y'=f(x,y) by f_y independent of y. Examples of linear equations and non-linear equations. Picard's theorem implies a linear DE has a unique solution. Main theorem on linear DE and explicit general solution.

References for linear DE: