Making Equations Separable Many differential equations that are not separable can be made separable by making a proper substitution. One example is the class of first-order equations with right-hand sides that are functions of the combination y/r (or r/ y1. Given such a DE

Called Euler-homogenous. Let v = y/t. By the product rule we deduce from y = vt that

So the equation becomes

Which separates into

Apply this method to solve the Euler homogenous Des and IVPs in problems. Plot sample solution on the direction field and discuss.
a.

b.

Transcribed Image Text:

dy=f(?) 4 dr = ' + ' dr. 4 dv , a dy y+ r dt dy y2+ 2