The Chain Rule & Implicit Differentiation · Superchallenge problem

A function $u(x,y)$ with continuous second partial derivatives satisfies $u_{xx} + 2u_{xy} - 3u_{yy} = 0$ on the whole plane, together with $u(x, 0) = \sin x$ and $u_y(x, 0) = 0$ for every $x$. Find $u\!\left(\pi, \dfrac{3\pi}{2}\right)$.