Solving the functional equation $f\left(y^2f(x)+x^2f(y)\right)=xy\big(f(x)+f(y)\big)$

Question

Problem: find all continuous functions $f:[0,+\infty)\to [0,+\infty)$ such that $$f\left(y^2f(x)+x^2f(y)\right)=xy\big(f(x)+f(y)\big),\;\forall x,y\in [0,+\infty)\text.$$