Find the $\displaystyle general solution$ of the equation $\displaystyle x+2y\sqrt{x^{2}+1}\frac{dy}{dx}=0$
I then separated the variables:
$\displaystyle
2ydy=\frac{-x}{\sqrt{x^2+1}}dx
$
I'm just struggling with the integration part.
Find the $\displaystyle general solution$ of the equation $\displaystyle x+2y\sqrt{x^{2}+1}\frac{dy}{dx}=0$
I then separated the variables:
$\displaystyle
2ydy=\frac{-x}{\sqrt{x^2+1}}dx
$
I'm just struggling with the integration part.
I'm assuming it's the RHS causing you trouble: $\displaystyle -\int \frac{x}{\sqrt{x^2+1}} \ dx$
Let: $\displaystyle {\color{red}u = x^2 + 1} \ \Rightarrow \ du = 2x \ dx \ \Leftrightarrow \ {\color{blue}\frac{1}{2}du = xdx}$
So: $\displaystyle -\int \frac{{\color{blue}x}}{\sqrt{{\color{red}x^2+1}}} \ {\color{blue}dx} = -{\color{blue}\frac{1}{2}} \int \frac{1}{\sqrt{{\color{red}u}}} {\color{blue}du} = -\frac{1}{2}\int u^{-\frac{1}{2}}du$