MAPLE LAB 2. [laptop projection] Hand Solution to L2.2. Graphic in L2.3. Interpretation of graphics in L2.4. PROBLEM 4.7-26. How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x) Outline of the general theory used to solve linear differential equations. Order of a DE and the dimension of the solution space. Euler's theorem. Finding solution atoms for a basis. BASIS. Definition of basis and span. Examples: Find a basis from a general solution formula. Bases and the pivot theorem. Equivalence of bases. A test for equivalent bases. DIMENSION. THEOREM. Two bases for a vector space V must have the same number of vectors. Examples: Last Frame Algorithm: Basis for a linear system Ax=0. Last frame algorithm and the vector general solution. Basis of solutions to a homogeneous system of linear algebraic equations. Bases and partial derivatives of the general solution on the invented symbols t1, t2, ... DE Example: y = c1 e^x + c2 e^{-x} is the general solution. What's the basis?

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EXAMPLE. The equation y'' 10y'=0. How to solve y'' + 10y' = 0 with chapter 1 methods. Midterm 1 problem 1(d). Idea: Let v=x'(t) to get a first order DE in v and a quadrature equation x'(t)=v(t). Solve the first order DE by the linear integrating factor method. Then insert the answer into x'(t)=v(t) and continue to solve for x(t) by quadrature. Vector space of functions: solution space of a differential equation. A basis for the solution space of y'' + 10y'=0. 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 THEOREM. It says that nth order equations have a solution space of dimension n. EULER'S THEOREM. It says 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 the formal replacement y ==> 1, y' ==> r, y'' ==> r^2. EXAMPLE. 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 vector space dimension theory, 1, exp(-10x) are a basis for the solution space of the differential equation. Then the general solution is y = c1 (1) + c2 (exp(-10x)).

Theory of Higher Order Constant Equations:Homogeneous and non-homogeneous structure. Superposition. Picard's Theorem. Solution space structure. Dimension of the solution set. Atoms. Definition of atom. Independence of atoms. Euler's theorem. Real roots Non-real roots [complex roots]. How to deal with conjugate pairs of factors (r-a-ib), (r-a+ib). 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 solution atoms. Examples: y''=0, y''+3y'+2y=0, y''+y'=0, y'''+y'=0.Second order equations.Homogeneous equation. Harmonic oscillator example y'' + y=0. Picard-Lindelof 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. 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 characteristic equation. How to solve any constant-coefficient homogeneous differential equation. Picard's Theorem for higher order DE and systems.

PROBLEM SESSION. Chapter 4 exercises.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, RLC circuit equation. harmonic oscillation,

How to construct solutions to 4.7-10,22,26. Questions answered on 4.3, 4.4, 4.5, 4.6, 4.7 problems. Survey of 4.3-4.4 problems. Illustration: How to do abstract independence arguments using vector packages, without looking inside the packages. Applications of the sample test and Wronskian test. How to use the pivot theorem to identify independent vectors from a list.

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. 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. Solved examples like the 5.1,5.2,5.3 problems. Solving a DE when the characteristic equation has complex roots. Higher order equations or order 3 and 4. 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: Applications. Damped and undamped motion.Theory of equations and 5.3-32. Problems discussed in class: all of 5.2, 5.3 and 5.4-20,34 spring-mass equation, LRC-circuit equation, Spring-mass DE and RLC-circuit DE derivations. Electrical-mechanical analogy. forced systems. harmonic oscillations, phase-amplitude conversions from the trig course. 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. Delayed: Beats. Pure resonance. Pendulum. Cafe door. Pet door. Over-damped, Critically-damped and Under-damped behavior, pseudoperiod.

- References: Sections 5.4, 5.6. Forced oscillations.

Lecture: Introduction to Laplace theory.Newton and Laplace: portraits of the Two Greats [slides]. Delayed: Method of quadrature. Comparison of Newton calculus and Laplace calculus. Direct Laplace transform == Laplace integral.