Solution
= Solution
Let $t=y/x$. The <product rule> and the <chain rule> give the <partial derivatives>
$$
u_x=\phi(t)+x\phi'(t)\left(-\frac y{x^2}\right)=\phi(t)-t\phi'(t),\qquad u_y=x\phi'(t)\frac1x=\phi'(t).
$$
Hence the terms containing $\phi'$ cancel:
$$
xu_x+yu_y=x\phi(t)-xt\phi'(t)+y\phi'(t)=x\phi(t)=\boxed{u}.
$$
This also follows from the <Euler theorem for homogeneous functions>, because $u(\lambda x,\lambda y)=\lambda u(x,y)$ for $\lambda>0$.