# Differential equation

• February 8th 2009, 06:44 PM
Konidias
Differential equation
Find the $general solution$ of the equation $x+2y\sqrt{x^{2}+1}\frac{dy}{dx}=0$

I then separated the variables:
$

2ydy=\frac{-x}{\sqrt{x^2+1}}dx
$

I'm just struggling with the integration part.
• February 8th 2009, 06:55 PM
o_O
I'm assuming it's the RHS causing you trouble: $-\int \frac{x}{\sqrt{x^2+1}} \ dx$

Let: ${\color{red}u = x^2 + 1} \ \Rightarrow \ du = 2x \ dx \ \Leftrightarrow \ {\color{blue}\frac{1}{2}du = xdx}$

So: $-\int \frac{{\color{blue}x}}{\sqrt{{\color{red}x^2+1}}} \ {\color{blue}dx} = -{\color{blue}\frac{1}{2}} \int \frac{1}{\sqrt{{\color{red}u}}} {\color{blue}du} = -\frac{1}{2}\int u^{-\frac{1}{2}}du$
• February 9th 2009, 01:02 AM
Konidias
I got $y^2=-\sqrt{x^2+1}+C$

Is this correct, and if so, how would you get it so its y= ?