<s>[INST] Non-linear Second Order Differential equation I want to ask for a hint in solving the following ODE,
$y'' (1+x^2) + y' *(x) = C$, 
where C is a constant. I've tried a couple of ways to manipulate this ODE; such as letting v = y'
$v' (1+x^2) + v *(x) = C$ 
but I can't see a way to solve the equation for y
 [/INST] $$
\left(1+x^2\right)v' + xv = C
$$
we can obtain
$$
v\sqrt{1+x^2} = C\int\frac{1}{\sqrt{1+x^2}}dx + \lambda_{1}
$$
or
$$
\frac{dy}{dx} = \frac{C}{\sqrt{1+x^2}}\int\frac{1}{\sqrt{1+x^2}}dx  + \frac{\lambda_{1}}{\sqrt{1+x^2}} = \frac{1}{\sqrt{1+x^2}}\left[C\sinh^{-1}(x) + \lambda_1\right]
$$
so
$$
y = \lambda_1\int \frac{1}{\sqrt{1+x^2}} + C\int \frac{1}{\sqrt{1+x^2}}\sinh^{-1}(x)
$$
the first integral evaluates to $\sinh^{-1}x$ the second can be evaluated as follows
$$
\int \frac{1}{\sqrt{1+x^2}}\sinh^{-1}(x) = \int \sinh^{-1}(x) \frac{d}{dx}\sinh^{-1}(x) dx = \frac{1}{2}\left(\sinh^{-1}x\right)^{2} + \lambda_2
$$
$$
y(x) = \lambda_1 \sinh^{-1}(x) + \frac{C}{2}\left(\sinh^{-1}x\right)^{2} + \lambda_3
$$
</s>