Notes

Slide Show

Outline

1	Non-Separable Equations Some Examples
2	Homogeneous Equations One definition of a first order homogeneous differential equation is that if the variables involved are x and y then it is an equation that can be written in the form
3	Make It Separable
4	Now Separable Equation (2) on the previous slide is separable. We now solve equation (2) and then replace v with y/x in our solution. Sometimes it may be easier to let v = x/y (this will also work). Sometimes it is easier to begin with the equation in differential form and make our substitutions in terms of differentials, i.e., M(x,y)dx + N(x,y)dy = 0 and dy = vdx + xdv.
5	Example
6	Solving
7	Initial Condition
8	Specific Initial condition
9	Graph The graph of the solution corresponding to the initial condition y(1) = 1 is shown in red on the next slide. The solution would not be meaningful beyond the vertical asymptote y = e. The direction field for this DE is shown in blue.
10
11	An Equation Of The Form
12	Equation (3) Becomes
13	Example
14	Example Continued
15	Exact Differential Equations
16	Total Differential
17	Exactness
18	Testing For Exactness
19	Necessary And Sufficient
20	Example—How To Find F
21	Finding F
22	Still Finding F
23	Finally Finding F
24	An Alternative
25	Another Option
26	Some Details
27	Solving The Equation
28	The Solution To The Equation
29	Initial Condition
30	Solving For y
31	Graphs
32
33
34
35
36	Zooming In On A Peak
37
38	Integrating Factors Some differential equations that are not exact can be made exact by multiplying each side of the equation by some function called an integrating factor. If I multiplied each side of the equation in the previous example by x the equation would no longer be exact. If I then multiplied each side of the new (non-exact) equation by 1/x (an integrating factor) the resulting equation would be exact.
39	Finding An Integrating Factor No integrating factor exists for most non-exact equations. An integrating factor that does exist but is a function of both variables, say u(x,y), may be next to impossible to find other then by a lucky guess. A positive valued integrating factor that is a function of only one of the variables, say u(x) or u(y), can be found if it exists.
40	An Integrating Factor, u(x)
41	Looking For u(x)
42	Looking For u(x)
43	Finding u(x)
44	An Integrating Factor u(y)
45	Example
46	Example Continued
47	Example Continued
48	Alternative
49	Animation Of Solutions
50	Solution Graph, y(0) = 1
51	Solution Graph, X-Zoom
52	Example: Sect 2.4 #35
53	2.4 #35 Continued
54	Solving For y, Initial Condition
55	Solution Graph
56	Linear Equations A first order linear differential equation is one that can be written in the following form:
57	Integrating Factor
58	Multiplying By u(x) Yields
59	Finding u(x)
60	To Solve A Linear Equation
61	Solving A Linear Equation Put the equation in the form y’ + Py = Q. Identify the functions P(x) and Q(x). Compute the integrating factor, u(x). You could then quickly solve the equation from the form given below as long as you can do the integration.
62	Example
63	Initial Condition
64
65	Animation
66
67	Return To DE Exploring Solutions to Sample Problems