Solving this ordinary differential equation

Question

Solve

$$\frac{dy}{dx} = 2\frac{y}{x} + \frac{x^3}{y} + x \tan\left(\frac{y}{x^2}\right).$$

I am unable to start this question.

Answer 1

Try $u=\frac{y}{x^2}$. This gives $xu'=\frac{1}{u} + \tan(u)$.

So $$\int du \, \frac{1}{\frac{1}{u} + \tan(u)}= \int du\, \frac{u \cos(u)}{u \sin(u) + \cos(u)}= \int \frac{dx}{x}.$$

This leads to $\log(u \sin(u)+ \cos(u))=\log(x) + \mathrm{const}$.

I'll leave the rest to you.