# ODE, seperation of variable

1. Sep 23, 2010

### fluidistic

1. The problem statement, all variables and given/known data
Verify that the following ODE can be reduced to an ODE of separable variables.
$$\frac{dy}{dx} =f(ax+by+c)$$ where a, b and c are constants.

2. The attempt at a solution
I think I must show that there exist functions g and h such that $$g(y)dy=h(x)dx$$.
I have that $$dy=f(ax+by+c) dx$$. I was at a loss. So I talked to a friend and he told me to write $$u=ax+by+c$$.
So I get $$dy=f(u)dx \Rightarrow y= \int f(u)dx=\frac{u-ax-c}{b}$$, $$y'=\frac{u'-a}{b}$$, $$y''=u''$$. I want to write $$f(u)$$ as $$\phi _1 (x) \phi _2 (y)$$ but I'm totally stuck.
I'd love a tip.
Thank you.

2. Sep 23, 2010

### LCKurtz

You have figured out that y' = (u'-a)/b and if you plug that in for y' you get

(u'-a)/b = f(u)

Write u' as du/dx and, remembering that a and b are constants, see if you can't get the u terms and the x terms on opposite sides.

3. Sep 23, 2010

### sachinism

ya lckurtz sums it up

also try looking here:http://www.tutorvista.com/math/separable-differential-equation" [Broken]

Last edited by a moderator: May 4, 2017
4. Sep 27, 2010

### fluidistic

Thank you guys, I got it.