Differential Equations – Day 4

Agenda:

  1. Homogeneous Equations
    & Functions
  2. Numerical Solutions
    to clip_image002
  3. Review of material
    in Chapter 1 & 2
  4. Models as well

Homogeneous Functions

If clip_image004 f is a homogeneous function of degree Alpha.

When alpha goes to 0, e.g. M and N are homogeneous of the same degree, use the substitution clip_image006.

clip_image008

clip_image010 becomes clip_image012

Factor and clip_image014 and clip_image016goes away. We can clean it up further by factoring again and moving things around.

clip_image018

clip_image020

clip_image022

clip_image024

Example:

clip_image026

clip_image028

clip_image030

clip_image032

clip_image034

clip_image036

clip_image038

Flow chart for solving thus far

  1. Is it separable? – Cool, integrate
  2. Is it linear? – Cool, use an integrating factor
  3. Is it exact? – Cool, solve it with integrating factor and or if its homogeneous cool
  4. Call Euler and use numerical method. Tangent and find with error.

clip_image040

clip_image042