Hello .

Question:

Solve the following ODE :

$\displaystyle y^2dx + (x^2+3y+4y^2)dy = 0$

Solution:

It is not Linear or Bernoulli in x or y.

It is not Exact.

It is not Separable.

It can not be converted it to an exact equation by the integrating factor.

I tried to substitute $\displaystyle t=y^2 \implies dy=\dfrac{dt}{2\sqrt{t}}$ to get:

$\displaystyle 2t^{3/2}dx+(x^2+3\sqrt{t}+4t)dt=0$

Which is ,again, I do not know what is its type!

Maybe it can be solved by inspection, but I can not figure it out.

Any help?