The question is as follows:
Consider the following partial differntial equation (PDE)
$2\frac{\partial^2u}{\partial x^2}+2\frac{\partial^2u}{\partial y^2} = u$
where $u=u(x,y)$ is the unknown function.
Define the following functions:
$u_1(x,y):=xy^2, u_2(x,y)=\sin(xy)$ and $u_3(x,y)=e^{\frac{1}{3}(x-y)}$
Which of these functions are solutions to the above PDE?
Any walkthroughs, description of methods, links to resources would be highly appreciated.
Same as you would an ordinary differential equation.
For each of $u_1,u_2,u_3$, do the following for this equation:
Calculate the relevant partial derivatives: that is, find $(u_1)_{xx},(u_1)_{yy},(u_2)_{xx},(u_2)_{yy},(u_3)_{xx},(u_3)_{yy}$
Substitute these values into the original PDE (which is $2u_{xx} + 2u_{yy} = u$): you're just picking $u = u_1$ when you use $u_1$ as the potential solution, and $u=u_2$ and $u=u_3$ in the other cases.
Substitute in the corresponding function $u_1$ or $u_2$ or $u_3$ for $u$.
Remember that function equality necessitates equality for all inputs for the functions. Thus, you should either ...
...perform manipulations algebraically to try and arrive at a true statement. For example, if you did manipulations and concluded with $1=1$ or whatever, then the two functions are equal (translating to the function you used being a solution).
...see if there's an input for which the two sides are not equal. For example, if I had, more simply, wanted to check if $x^2 = x^9$, I could see that if I put in $x=1/2$ I get two very different values. Thus the functions are not equal (which would translate into $u_1$ or $u_2$ or $u_3$, whichever you used, not being a valid solution).