Answer
$u_{1,x}=-\sin x \cosh y,\ u_{1,xx}=-\cos x \cosh y$
$u_{1,y}=\cos x \sinh y,\ u_{1,yy}=\cos x \cosh y$
$u_{1,xx}+u_{1,yy}=0$
$u_{2,x}=\dfrac{2x}{x^2+y^2},\ u_{2,xx}=\dfrac{-2(x^2-y^2)}{x^2+y^2}$
$u_{2,y}=\dfrac{2y}{x^2+y^2},\ u_{2,yy}=\dfrac{2(x^2-y^2)}{x^2+y^2}$
$u_{2,xx}+u_{2,yy}=0$
