You can solve the equation $x{dy\over dx}=4y$ using separation of variables: $\tag{1} {1\over y}\,dy={4\over x}\,dx. $ We want to avoid division by zero; so, in solving $(1)$, we consider the cases where $x<0$ and $x>0$. Integration of $(1)$ shows that for $k$ a constant, $y=k x^4$ is a solution for both the domain $x>0$ and the domain $x<0$.
One may then verify that the solutions of $(1)$ over $\Bbb R$ have the form $\tag{2} f(x)=\cases{cx^4,&$x\ge0$\cr d x^4,&$x\le 0$ } $ where $c$ and $d$ are constants (in particular, one can (and should) show such an $f$ is differentiable at $0$ with $f'(0)=0$).
Note that any such $f$ is a linear combination of the independent functions obtained by taking $(c=1, d=0)$ and $(c=0,d=1)$ in $(2)$.