Ordinary Differential Equations – How to Solve the Differential Equation $ \\frac{\\mathrm d^2y}{\\mathrm dx^2}(y\\frac{\\mathrm dx}{\\mathrm dy}+x) = -A^2\\frac{\\mathrm dy}{\\mathrm dx}$

Question

I came across this differential equation while investigating how the concentration of a fluid varies with position $ x $ ($ A $ is a constant).

I tried to solve this by using a substitution, but I was unable to actually solve the equation for y. Wolfram Alpha did not come up with anything useful, either.

Does anyone have any idea how to solve this? I would greatly appreciate if anyone could help me with this.

Answer 1

Note that the equation is unchanged if you double $y$, and is also unchanged if you double $x$.
So substitute $y=\exp(w)$ and $x=\exp(t)$. The result will involve $dw/dt$ and $d^2w/dt^2$ but not $w$ because the equation is unchanged if you add a constant to $w$. It won't involve $t$ explicitly because the equation is unchanged if you add a constant to $t$.
The result is a first-order separable DE in $v(t)=dw/dt$