I’m trying to solve the IVP $y’=\dfrac{f(y)}{1+x^4}$ with the initial condition $y(0)=(0)$. Here $$f(y)=|y|\ln|y|$$ for $y \neq 0$ and $f(0)=0$.
Using separation of variables I arrive at $$\frac{dy}{dx}=\dfrac{f(y)}{1+x^4} \iff \int \dfrac{dy}{f(y)}=\int \dfrac{1}{1+x^4}dx.$$ Is there any way to calculate $$\int \dfrac{1}{f(y)}dy?$$ Or is there a simpler way to solve this ODE?
Note that $$\int \dfrac{1}{f(y)}dy= \int \dfrac{1}{|y|\ln |y|}dy=\text{sgn}(y)\ln (\ln |y|) $$