# 2250-4 12:55pm Lecture Record Week 2 S2010

Last Modified: January 24, 2010, 09:43 MST.    Today: August 16, 2018, 00:43 MDT.

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

```Topics delayed from Week 1.
implicit solution ln|y|=2x+c for y'=2y
explicit solution y = C exp(2x) for y'=2y
Troubles with explicit solutions of 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
Example 1: Show that y'=x+y is not separable using TEST I or II
(partial derivative tests).
Example 2: Find a 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].
```
```Review Topics
Drill: Direction fields.
Picard and Peano Theorems.
We draw threaded solutions from some dot in the graphic. How
do we choose the dots? What do they represent?
What does dy/dx=f(x,y), y(x0)=y0 have to do with threaded
curves?
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].
```

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

```Review and Drill 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.
```
```Lecture on Section 1.5
Theory of linear DE y'=-p(x)y+q(x).
Integrating factor W=e^Q(x), Q(x) = int( p(x),x)
(Wy)'/W, the fraction that replaces two-termed expression y'+py.
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)

Linear integrating factor method 1.5
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.
```

## 20-21 Jan: Murphy and Cox

Present problems 2, 3 of the midterm 1 sample [f2009 midterm 1 key].
Exam 1 date is 17 Feb 1-3pm in WEB 104 or 18 Feb 6:50am in JTB 140. Other exam times were pre-set by agreement at the start of the semester, on an individual basis. The plan was created to provide extra time to write the exam, which is designed for 50 min.
Sample Exam: Exam 1 key from F2009. See also S2009, Exam 1.
HTML: 2250 midterm exam samples S2010 (15.6 K, html, 16 May 2010)
Questions on textbook sections 1.3, 1.4.
Review and drill Ch1.

## 21 Jan: Linear Applications. Section 1.5

Collect first package of dailies in class: 1.2, 1.3.
```Review and Drill
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
Superposition for y'+p(x)y=0.
Superposition for y'+p(x)y=q(x).
```
```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.
```

## 21 Jan: Linear Applications 1.5. Autonomous systems section 2.1

``` Drill
Section 1.5: Three linear examples: y'+(1/x)y=1, y'+y=x, y'+2y=1.
Methods for solving first order equations:
Linear integrating factor method,
Superposition + equilibrium solution for
constant-coefficient linear,
Variables Separable method
Equilibrium solutions from G(y)=0 and
Non-equilibrium solutions from G(y) nonzero.
```
```Lecture on 2.1
Theory of autonomous DE y'=f(y),
Picard's theorem and non-crossing of solutions.
Direction fields and translation of solutions
No direction field, just the phase line diagram.
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
Separation of variable solutions with partial fractions.
```
Midterm 1 sample exam is the F2009 exam, found at the course web site.
HTML: 2250 midterm exam samples S2010 (15.6 K, html, 16 May 2010)
To date, Murphy and Cox have covered problems 1,2,3 in the exam review sessions on Wed-Thu.