# 2250 8:05am Lectures Week 2 S2015

Last Modified: January 25, 2015, 12:53 MST.    Today: September 23, 2018, 14:02 MDT.
```Topics
Sections 1.5, 2.1, 2.2, 2.3
The textbook topics, definitions, examples and theoremsEdwards-Penney 1.2, 1.3, 1.4, 1.5 (14.6 K, txt, 12 Jan 2015)Edwards-Penney 2.1, 2.2, 2.3 (15.5 K, txt, 17 Dec 2014)Examples: Week 2 (290.4 K, png, 26 Jan 2015)```

### Wednesday: Solving Linear DE. Section 1.5.

```Lecture on Section 1.5
We will study linear DE y'=-p(x)y+q(x).
Classification of y'=f(x,y)
quadrature [Q], separable [S], linear [L].
Venn diagram of classes Q, S, L.
Examples of various differential equation types.
Test for linear (f_y indep of y)
Test for not separable (f_y/f depends on x ==> not sep)
Section 1.4. Separable DE. Review and Drill, as time allows.
Variables separable method.
Finding F and G in a separable equation y'=F(x)G(y)
Equilibrium solutions from G(y)=0 and
Non-equilibrium solutions from G(y) nonzero.
Does f(x,0)=F(x)G(0) determine F(x)? Explain.
Method of Quadrature: When to use it.
Discuss remaining exercises 1.4-6,12,18.
Problem Notes Chapter 1 at the web site.
Detailed derivations for 1.4-6
y' = 3 sqrt(-x) sqrt(-y)  on quadrant 3, x<0, y<0
y' = 3 sqrt(x) sqrt(y)  on quadrant 1, x>0, y>0
Constant solution or Equilibrium solution
Found by substitution of y=c into the DE y'=3 sqrt(xy)
Ans: y=0 is an equilibrium solution
Non-constant solution or Non-equilibrium solution
Found from y'=F(x)G(y) by division by G(y),
followed by the method of quadrature.
y = ( x^(3/2)+c)^2
y = - ((-x)^(3/2)+c)^2
List of 3 solutions cannot be reduced in number
How to test separable and non-separable equations
Theorem. If f_y/f depends on x, then y'=f(x,y) is not separable
Theorem. If f_x/f depends on y, then y'=f(x,y) is not separable
Theorem. If y'=f(x,y) is separable, then f(x,y)=F(x)G(y) is
the separation, where F and G are defined by the formulas
F(x) = f(x,y0)/f(x0,y0)
G(y) = f(x0,y).
The invented point (x0,y0) may be chosen conveniently,
subject to f(x0,y0) nonzero.
Section 1.5. Linear integrating factor method
References for linear DE: Slides: Linear integrating factor method (126.0 K, pdf, 15 Dec 2014)Manuscript: Applications of linear DE, brine tanks, home heating and cooling (484.2 K, pdf, 16 Jan 2014)Manuscript: 1st Order Linear DE part I. Integrating Factor Method, Superposition (303.6 K, pdf, 16 Jan 2014)Manuscript: 1st Order Linear DE part II. Variation of Parameters, Undetermined Coefficients (238.7 K, pdf, 16 Jan 2014)Manuscript: Kinetics, Newton's Models (343.8 K, pdf, 16 Jan 2014)Transparencies: Linear integrating factor method, exercises 1.5-3,5,11,33. Brine mixing (375.0 K, pdf, 29 Jan 2006)Text: How to do a maple answer check for y'=y+2x (0.3 K, txt, 07 Jan 2014)
Linear Differential Equation y'+p(x)y=q(x)
Section 1.5
Definition: Linear DE y'+p(x)y=q(x)
Test: y'=f(x,y) is linear if and only if the partial
derivative f_y is independent of y.
Examples:
Testing linear DE y'=f(x,y) by f_y independent of y.
Classifying linear equations and non-linear equations.
Picard's theorem implies a linear DE has a unique solution.
We don't check for equilibrium solutions or exceptions.
THEOREM. A linear DE has an explicit general solution (no implicit equations!).
Def. Integrating factor W=exp(Q(x)), where Q(x) = int(p(x),x)
THEOREM. The integrating factor fraction (Wy)'/W replaces the
two-termed expression y'+py.
Application Examples: y'+2y=1 and y'+y=e^x.
ALGORITHM.
How to solve a linear differential equation
Test the DE for linear
Identify p(x), q(x) in the standard form y'+py=q.
Determine an integrating factor W(x)=exp(int(p(x),x))
Replace y'+py in the standard form y'+py=q by the quotient
(Wy)' / W
and then clear fractions to get the quadrature equation
(Wy)' = qW
Solve by the method of quadrature.
Divide by W to find an explicit solution y(x).
Three linear examples: y'+(1/x)y=1, y'+y=e^x, y'+2y=1.
Superposition Theory
Superposition for y'+p(x)y=0.
Superposition for y'+p(x)y=q(x)
Methods for solving first order linear equations:
Failsafe: Linear integrating factor method,
Shortcut: Superposition + equilibrium solution for
constant-coefficient linear DE
Example. The fastest way to solve y'+2y=1 is the Shortcut.
Drill: Week 2 examples distributed in class,
Example: y' + 2y = 6.TEXT: Examples 1st order linear DE, integrating factor methods (4.2 K, txt, 14 Dec 2014)
Solved in class y'+3y=6, y'+y=e^x, and homogeneous equations
like y'+3y=0, y'+2y=0.
Solve for equilibrium solutions in strange examples like
2y' + Pi y = e^2.
Examples and Applications
Growth-Decay model y'=ky and its algebraic model y=y(0)exp(kx).
Pharmacokinetics of drug transport [PK models, ibuprofen]
Pollution models.
Three lake pollution model [Erie, Huron, Ontario].
Brine tanks.
One-tank model.
Two-tank and three-tank models.
Chemical engineering example, 3 tanks.
Recycled brine tanks and limits of chapter 1 methods.
Linear cascades and how to solve them.
Method 1: Linear integrating factor method.
Method 2: Superposition and equilibrium solutions for
constant-coefficient y'+py=q. Uses the shortcut for
homogeneous DE y'+py=0.
LR and RC circuits. Ref: EPbvp 3.7
Basic information on LR, RC and LC circuits.LR circuit HereRC circuit HereLC circuit Here
LR Circuit
LI' + RI = E
RC Circuit
RQ' + Q/C = E
Method: Linear integrating factor method, usually the shortcut
for constant equations.

Problem 1.5-34
The expected model is
x'=1/4-x/16,
x(0)=20,
using units of days and millions of cubic feet.
Model Derivation
Law:  x'=input rate - output rate.
Definition:  concentration == amt/volume.
Example: 200 grams of salt in 15 liters brine, conc=(0.2/15)kg/liter
Use of percentages
0.25% concentration means 0.25/100 concentration
References for linear applicationsManuscript: Applications of linear DE (484.2 K, pdf, 16 Jan 2014)Slides: Brink tanks (95.3 K, pdf, 03 Mar 2012)Slides: Home heating (99.3 K, pdf, 09 Apr 2014)```

### Wed-Fri: Linear DE. Autonomous systems Section 2.1

```Introduction to Ch 2 topics
2.1, 2.2: Autonomous DE y'=f(y)
Solution of the Verhulst DE y'=(a-by)y
2.3: Newton models, Jules Verne problem
2.4,2.5.26: Numerical solutions of DE.
No exercises, but a numerical project based on EPH exercises.
Solution of various exponential modelsManuscript: Exponential Application Library (340.2 K, pdf, 21 Jan 2015)
Drill on 1.5 Problems, as time allows
There are two special methods for solving y'+py=q
If p,q are constant then use the SUPERPOSITION METHOD
y = y_p + y_h
y_p = an equilibrium solution (set y'=0, solve for y)
y_h = constant divided by the integrating factor
If one of p or q depends on x, then use the STANDARD METHOD
Replace the LHS, which is y'+p(x)y, by the integrating
factor quotient (Wy)'/W, where W=exp(int p(x)dx)) is the
integrating factor.
Cross-multiply by W to clear fractions. Then apply the
method of quadrature.  Slides: Linear integrating factor method (126.0 K, pdf, 15 Dec 2014)Manuscript: Linear DE part I. Integrating Factor Method (303.6 K, pdf, 16 Jan 2014)
General Verhulst DE, Section 2.1
Solving y'=(a-by)y by a substitution
Let u=y/(a-by).
Then substitution into the DE gives u'=au
Solve u'=au to get u=u0 exp(ax).
Back-substitute u(x) into u=y/(a-by), then solve for y.
Solving y'=(a-by)y by partial fractions
Divide the DE by (a-by)y
Find the constants in the partial fractions on the left.
a y0
y(x) = --------------------------
b y0 + (a - b y0) exp(-ax)
where y0=y(0)=initial population size.
Review of Partial fractions.
DEFINITION: partial fraction=constant/polynomial with exactly one root
THEOREM: P(x)/q(x) = a sum of partial fractions
Finding the coefficients.
Method of sampling
clear fractions, substitute samples, solve for A,B, ...
Method of atoms
clear fractions, multiply out and match powers, solve for A,B,...
Heaviside's cover-up method
partially clear fraction, substitute root, find one constant
References on partial fractionsSlides: Partial Fraction Theory (148.6 K, pdf, 14 Dec 2014)Manuscript: Heaviside coverup partial fraction method (290.2 K, pdf, 07 Jan 2014)Manuscript: Heaviside's method and Laplace theory (352.3 K, pdf, 07 Jan 2014)
Separation of variables and partial fractions
Exercise solution problem 2.1-8
The equation y'=7y(y-13), y(0)=17
F(x) = 7, G(y) = y(y-13)
Separated form y'/G(y) = F(x)
Answer check using the Verhulst solution
P(t) = a P_0/(b P_0 + (a-b P_0) exp(-at))
Separation of variables details.
Partial fraction details for 1/((u(u-13)) = A/u + B/(u-13)
```

### Wed-Fri: Autonomous Differential Equations and Phase Diagrams. Section 2.2

```Lecture on 2.2:
Theory of autonomous DE y'=f(y)
Picard's theorem and non-crossing of solutions.
Direction fields and translation of solutions
No direction field is needed to draw solution curves
We throw out the threaded solution rule used in chapter 1,
replace it by two rules from calculus and a theorem:
1. If y'(x)>0, then y(x) increases.
2. If y'(x)<0, then y(x) decreases.
THEOREM. For y'=f(y), a threaded solution starting with
y'(0)>0 must satisfy y'(x)>0 for x>0. A similar result
holds for y'(0)<0.
==> Stopped here on Friday
Definition: phase line diagram, phase diagram,
Calculus tools: f'(x) pos/neg ==> increasing/decreasing
DE tool: solutions don't cross
Maple tools for production work.
Stability theory of autonomous DE y'=f(y)
Stability of equilibrium solutions.
Stable and unstable classification of equilibrium solutions.
funnel, spout, node,
How to construct Phase line diagrams
How to make a phase diagram graphic
Inventing a graph window
Invention of the grid points
Using the phase line diagram to make the graphic
calculus tools
DE tools
References for 2.1, 2.2, 2.3. Includes the rabbit problem, partial
fraction examples, phase diagram illustrations.Slides: Autonomous DE (101.8 K, pdf, 14 Jan 2014)Manuscript: Verhulst logistic equation (115.5 K, pdf, 02 Oct 2009)Manuscript: fish farming (384.5 K, pdf, 16 Jan 2014)Manuscript: Phase Line and Bifurcation Diagrams. Includes Stability, Funnel, Spout, and bifurcation (765.5 K, pdf, 14 Jan 2014)Text: Problem notes Chapter 2 (8.9 K, txt, 28 Jan 2015)Transparencies: ch2 sections 1,2,3: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (357.6 K, pdf, 29 Jan 2006)html: Problem notes S2015 (2.8 K, html, 01 Jan 2015)
```

### Thursday: Ziwen Zhu

```Second Lab project.
```