Pregunta: Solve the problem ∂x2∂2u+∂y2∂2u=0,0

Solve the problem $\begin{array}{l}\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0,0<x<\pi ,0<y<\infty \\ u(0,y)=0,u(\pi ,y)=e^{-y},0<y<\infty \\ u_y(x,0)=\sin (x),0<x<\pi \end{array}$ Which of the following is/are true: (I) Apply Fourier cosine transform with respect to $X$ (II) The transformed equation is $\frac{d^{2}U(x,\alpha )}{d{x}^2}-{\alpha }^{2}U(x,\alpha )-u_y(x,0)=0$ (III) $U(x,\alpha )=c_1\cosh (\alpha x)+c_2\sinh (\alpha x)-\frac{1}{\left(1+{\alpha }^2\right)}\sin (x)$ is the solutic