= Solution
The <chain rule> gives $\partial_x=\partial_u+\partial_v$ and $\partial_t=\partial_u-\partial_v$. Hence $\partial_x^2-\partial_t^2=4\partial_u\partial_v$, and the forced <wave equation> becomes $y_{uv}=1$. Integrating twice gives $y=uv+F(u)+G(v)$. At $t=0$ the initial conditions are
$$
x^2+F(x)+G(x)=\sin x,\qquad F'(x)-G'(x)=0.
$$
Thus $F-G$ is constant, which cancels from the final expression. Combining the remaining sum yields
$$
y=uv+\tfrac12(\sin u+\sin v-u^2-v^2)
=\boxed{\sin x\cos t-2t^2.}
$$
Differentiation checks the forcing and both initial conditions.
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