We have 2 problems:
1)
$$U_{xx}(x,y)+U_{yy}(x,y)=f(x,y):=-\pi^2\sin{(\pi x)}\sin{(\pi y)}$$
The conditions are
$$1=U(0,y)=U(1,y)=U(x,0)=U(x,1)$$
2)
$$U_{xx}(x,y)+U_{yy}(x,y)-U(x,y)=f(x,y):=-\pi^2\sin{(\pi x)}\sin{(\pi y)}$$
The conditions are
$$1=U(0,y)=U(1,y)=U(x,0)=U(x,1)$$
Can someone give me the solutions?
I know, it is usual in this site that the students have tried the problems by themselves and talk where they have difficulties. I am not specially good at PDE because I have not studied it yet, but I need the solutions of 1) and 2) with urgency. I usually tried the problem by myself before asking (you can check my other questions)
Thank you very much!
The RHS of your two problems imply the solution.
We can guess the solution have the form $u=c \sin(\pi x)\sin(\pi y)+1$ for these two problems which $c$ is a constant to be determined. Plug this solution into the PDEs and solve the $c$, we have $u=1/2\sin(\pi x)\sin(\pi y)+1$ for 1) and $u=\frac{\pi^2}{2\pi^2+1}\sin(\pi x)\sin(\pi y)+1$ for 2).