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

Solve the problem $\begin{array}{l}\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0,0<x<\infty ,0<y<5\\ u(x,0)=0,u(x,5)=e^{-x},0<y<5\\ u_x(0,y)=\sin \left(\frac{\pi y}{5}\right),0<x<\infty \end{array}$ Which of the following is/are true: (I) Apply Fourier cosine transform with respect to $x$ (II) The transformed equation is $\frac{d^{2}U(\alpha ,y)}{d{y}^2}-{\alpha }^{2}U(\alpha ,y)+u_x(0,y)=0$ (III) $U(\alpha ,y)=c_1\cosh (\alpha y)+c_2\sinh (\alpha y)-\frac{1}{\left(\frac{{\pi }^2}{25}+{\alpha }^2\right)}\sin \left(\frac{\pi y}{5}\right)$ is the solution of the correct transformed equation. All are correct No one is correct (I) and (II) only (II) and (III) only (I) and (III) only