2250-1 7:30am Lecture Record Week 2 S2012

Last Modified: January 26, 2012, 19:51 MST.    Today: July 17, 2018, 17:24 MDT.

17 Jan: Theory of Linear First Order Differential Equations. Section 1.5.

``` Theory of separable equations section 1.4.
Separation test:
Define 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.
Non-Separable Test
TEST I. f_x/f depends on y ==> y'=f(x,y) not separable
TEST II. f_y/f depends on x ==> y'=f(x,y) not separable
Review: Basic theory of y'=F(x)G(y):
y(x) = H^(-1)( C1 + int(F)),
H(u)=int(1/G,u0..u).
Solutions y=constant are called equilibrium solutions.
Find them using G(c)=0.
Non-equilibrium solutions arise from y'/G(y)=F(x) and a
```
```Implicit and explicit solutions.
Discussion of answer checks for implicit solutions and also
explicit solutions.
Exercise 1.4-6: Trouble with explicit solutions of y'= 3 sqrt(xy)
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].
```
```Answer Checks and Key Examples.
implicit solution ln|y|=2x+c for y'=2y
explicit solution y = C exp(2x) for y'=2y
Answer check for y'= 3 sqrt(xy) [1.4-6].
Key Examples
Separable DE with no equilibrium solutions.
Separable DE with finitely many 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
```
```Lecture on Section 1.5
Theory of 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 types.
Test for linear (f_y indep of y)
Test for not separable (f_y/f depends on x ==> not sep)
Finding F and G in a separable equation y'=F(x)G(y)
```

18 Jan: Linear Applications. Section 1.5

```Review and Drill Section 1.4
Variables separable method.
Discuss remaining exercises 1.4-6,12,18.
Problem Notes 1.4 at the web site.
Equilibrium solutions and how to find them.
```
```Review and Drill
Variables Separable method
Equilibrium solutions from G(y)=0 and
Non-equilibrium solutions from G(y) nonzero.
```
```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
Equilibrium solution
Found by substitution of y=c into the DE y'=3 sqrt(xy)
Ans: y=0 is an equilibrium solution
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 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.
Linear integrating factor method 1.5
Def. Integrating factor W=e^Q(x), Q(x) = int( p(x),x)
(Wy)'/W, the fraction that replaces two-termed expression y'+py.
Application to y'+2y=1 and y'+y=e^x.
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.
Main theorem on linear DE and explicit general solution.
```
```Linear Differential Equation y'+p(x)y=q(x)
Section 1.5
Definition: Linear DE
Test: y'=f(x,y) is linear if and only if the partial
derivative f_y is independent of y.
Algorithm
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)dx))
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.
Two Methods for solving first order equations:
Linear integrating factor method,
Superposition + equilibrium solution for
constant-coefficient linear,
```

19 Jan: Patrick B.

```Maple lab 1: quadratics, partial derivatives.
Present problem 1 from the midterm 1 sample [S2010 midterm 1 key].
Exam 1 date is in the syllabus and also the online due dates page.
Questions on textbook sections 1.3, 1.4.
Review and drill Ch1.
Sample Exam: Exam 1 key from F2010. See also S2010, Exam 1.
```

HTML: 2250 midterm exam samples S2012 (18.9 K, html, 02 May 2012)

20 Jan: Solving Linear DE. Start Ch 2.

```Superposition Theory
Superposition for y'+p(x)y=0.
Superposition for y'+p(x)y=q(x)
A faster way to solve y'+2y=1
```
```Drill Section 1.5
Three linear examples: y'+(1/x)y=1, y'+y=e^x, y'+2y=1.
Methods for solving first order equations:
Linear integrating factor method,
Superposition + equilibrium solution for
constant-coefficient linear DE
Drill: worksheet distributed in class, for the example
y' + 2y = 6. Solved in class y'+3y=6, y'+y=e^x, and several
homogeneous equations like y'+3y=0, y'+2y=0. Solved for
equilibrium solutions in more complicated examples like
2y' + Pi y = e^2.
```
```Problem 1.5-34
The expected model is
x'=1/4-x/16,
x(0)=20,
using units of millions of cubic feet.
Model Derivation
Law:  x'=input rate - output rate.
Definition:  concentration == amt/volume.
Use of percentages
0.25% concentration means 0.25/100 concentration
```
```Introduction to Ch 2 topics
Autonoumus DE y'=f(y)
Solution of the Verhulst DE y'=(a-by)y
Numerical solutions of DE. No exercises, but:
Maple lab 3
Maple lab 4
```
Midterm 1 sample exam is the F2010 exam, found at the course web site.
HTML: 2250 midterm exam samples S2012 (18.9 K, html, 02 May 2012)