What course is this for? It looks like a straightforward separble equation. If you let then the equation becomes XT"= X"T or T"/T= X"/X. Since the left hand side is a function of t only and the right hand side is a function of x only, each side must be a constant. Calling that constant , we have or and or .

However, there is a problem with the additional conditionsl. Since this equation is second order in both x and t, you needtwoboundary andtwoinitial conditions. You have only one of each.