Which method to solve $2xyy'=x^2+3y^2$?

Question

As the title states I'm trying to solve $2xyy'=x^2+3y^2$. I have explored the methods such as integrating factor, separable equations, and exact equations. The equation above doesn't seem to give way to any of these methods.

Note: I have never learned the method of exact equations where you multiply by a factor that turns the equation into an exact equation. That could be it but I wanted to run it by some others to see if maybe I missed something obvious.

Answer 1

$$2xyy'=x^2+3y^2$$ $$x(y^2)'=x^2+3y^2$$ Substitute $z=y^2$ $$ \implies xz'-3z=x^2$$ $$ x^3z'-3x^2z=x^4$$ $$\left(\frac z {x^3}\right)'=\frac 1 {x^2}$$ Simply integrate

Answer 2

After noticing that the exponents line up correctly, we can rewrite this as

$$2y' = \frac{x}{y} + 3 \frac{y}{x}$$

which suggests the standard substitution $v = y/x$.

Answer 3

The equation is homogeneous, so you can use the substitution $u=y/x$, $y'=u'x+u$.