# Thread: Easy First Order ODE

1. ## Easy First Order ODE

Solve,

$y\frac{dy}{dx} = 1+2y$

I know this is easy but I have gone blank its seperable but I had trouble integrating for some reason.

thanks

2. $\int\frac{y}{1+2y}dy\;=\;\frac{y}{2}-\frac{1}{4}ln|1+2y|+C$?

3. Originally Posted by Jason Bourne
Solve,

$y\frac{dy}{dx} = 1+2y$

I know this is easy but I have gone blank its seperable but I had trouble integrating for some reason.

thanks
Notice that
$\frac{y}{1 + 2y} ~dy = dx$

Let $u = 2y + 1 \implies du = 2dy$ so

$\frac{\frac{1}{2}(u - 1) }{u}~\frac{1}{2}du = dx$

$\frac{1}{4}\frac{u - 1}{u}~du = dx$

$\frac{1}{4} \left ( 1 - \frac{1}{u} \right ) du = dx$

-Dan