<s>[INST] How do I solve this differential equation? $y'=\frac{y}{x+y^3}$ So, hey.
The equation is $y'=\frac{y}{x+y^3}$
So, I get something like this: $y'\left(x+y^3 \right)-y=0$, which I can't actually solve. I must admit I am slightly confused how to attack this one.
What should I do? 
 [/INST] $y'=\frac{y}{x+y^3}\Leftrightarrow(x+y^3)dy=ydx$.
Case 1: $y=0$.
Case 2: $y\neq0$, then $$\frac{1}{y^2}[(x+y^3)dy-ydx]=ydy-xd\frac{1}{y}-\frac{1}{y}dx=d(\frac{1}{2}y^2-\frac{x}{y})$$
$\frac{x}{y}-\frac{1}{2}y^2=C$.
The solution to the ODE is $x=\frac{1}{2}y^3+Cy$.
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