<s>[INST] Solve $y' = x^4y+x^4y^4$ Solve the differential equation $$y' = x^4y+x^4y^4.$$
I'm not sure how to deal with the $x^4y^4$ term. So far I have only encountered differential equations where the exponent of $y$ was at most one.
Could someone please share a general strategy/hint?
(This is test preparation, not homework.)
 [/INST] $$\frac{y'}{y+y^4}=x^4\tag{1}$$
and since
$$\int\frac{dz}{z+z^4}=\frac{1}{3}\log\frac{z^3}{1+z^3}\tag{2}$$
by partial fraction decomposition, by integrating both terms in $(1)$ we get:
$$\log\left(1+\frac{1}{y^3}\right)=-\frac{3}{5}x^5+C,\tag{3}$$
hence:
$$ 1+\frac{1}{y^3}= K e^{-\frac{3}{5}x^5},$$

$$ y= \frac{1}{\sqrt[3]{K e^{-\frac{3}{5}x^5}-1}}.\tag{4}$$

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