Beyond Linear DEs

Calculus Level 2

Given that $y=f(x), \frac{dy}{dx}=x^{-4}+y^2,$ and $f\big(\frac{1}{\pi}\big)=-\pi,$ find $f\big(\frac{4}{3\pi}\big).$

\dfrac{3}{16}\pi\left(3\pi+4\right)

\dfrac{3}{16}\pi\left(4-3\pi\right)

-\dfrac{3}{16}\pi\left(3\pi+4\right)

\dfrac{3}{16}\pi\left(3\pi-4\right)