Question

Quick question, is there a way to know if when you are dealing with a homogeneous equation instead of an exact equation or vice versa when solving for a DE?

Answer 1

A homogenous equation can be exact from what I researched in the last couple minutes

Answer 2

Is this just a general question or is there a specific DE

Answer 3

This is mostly a general question, i was wondering because i wanted to know how i could approach problems more efficiently.

Answer 4

I was just studying this topic, so here goes:

Form of homogeneous DE: M(x,y) + N(x,y) dy/dx = 0, where M(x,y) and N(x,y) are functions of x and/or y. Both M(x,y) and N(x,y) are such that f(cx,cy) = c^n f(x,y). {Both M and N are represented by f in this case}. Here, n is the degree of the homogeneous function.
Method of solution: let y=ux.
EXAMPLE: \[(x^2 + y^2) + (x^2 -xy)\frac{ dy }{ dx } = 0\]

Form of exact DE: M(x,y) + N(x,y) dy/dx = 0, where d/dy M(x,y) = d/dx N(x,y).
Method of solution: Basically, LHS = RHS =>\[\int\limits M(x,y) dx = \int\limits N(x,y) dy\]