= Solution
The convective acceleration is
$$
u u_x+v u_y=\sin x\cos x,\qquad
u v_x+v v_y=-\sin y\cos y.
$$
Steady <Euler equations> therefore give $p_x=-\rho\sin x\cos x$ and $p_y=\rho\sin y\cos y$. Integrating these compatible equations yields
$$
\boxed{p(x,y)=p_0-\frac\rho2\sin^2x+\frac\rho2\sin^2y},
$$
with arbitrary additive constant $p_0$. As a check, $p/\rho+(u^2+v^2)/2=p_0/\rho+1/2-\psi^2$, indeed constant on each <streamline>. This is a <Steady Euler flow with vorticity proportional to its stream function>.
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