TITLE: How do I solve this specific ordinary differential equation?
QUESTION [5 upvotes]: I have no clue how to solve this ode , I've been watching videos, searching online, but I just can't, someone please help me.
$$2xyy' = x^2 + xy$$

REPLY [3 votes]: It's separable if you substitute $y=tx$
and $y'=t+xt'$
$$2y'=\frac x y+1$$
$$2(t+xt')=\frac 1 t+1$$
$$2xt'=\frac 1 t+1-2t$$
$$ 2xdt=(\frac 1 t +1 -2t)dx$$
$$ \int  \frac 1 {2x}dx=\int\frac {tdt} { 1+t-2t^2}$$
$$ \frac {\ln(x)} {2}+K=\int\frac {tdt} {(1-t)(1+2t)}$$