# First-order nonlinear ordinary differential equation

• Sep 12th 2011, 08:14 PM
mafra
First-order nonlinear ordinary differential equation
How do I solve this one?

(2x - y + 4)dy + (x - 2y + 5)dx = 0
• Sep 12th 2011, 08:20 PM
alexmahone
Re: First-order nonlinear ordinary differential equation
Quote:

Originally Posted by mafra
How do I solve this one?

(2x - y + 4)dy + (x - 2y + 5)dx = 0

$\frac{dy}{dx}=\frac{2y-x-5}{2x-y+4}$

Let $x=u+h$ and $y=v+h$, where $h$ and $k$ are constants that are to be determined.

$dx=du$ and $dy=dv$

We want $\frac{dv}{du}=\frac{2v-u}{2u-v}$, so that we have a homogeneous equation, that can be easily solved.

Find $h$ and $k$ by setting $2y-x-5=2v-u$ and $2x-y+4=2u-v+4$.