How can I solve my differential equation with Mathematica?

[Hertz]

Hi, I can't quite figure out what I'm doing wrong here. I tried restarting mathematica as well which is what fixed my previous problem

Clear[eqn, soln, u, x, y]
eqn = D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] == 0;
soln = NDSolve[{eqn, u[x, 0] == Sin[x], u[0, y] == 0, u[1, y] == 0}, 
  u[x, y], {x, 0, 1}, {y, 0, 1}]

and I'm getting the error:

NDSolve::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable. >>

 

[Bill Simpson]

There are rules about how you can and can't specify conditions when trying to solve differential equations and the wording of the error message appears to fairly clearly state that you have violated that rule. There is an entire text published on only how to solve differential equations with Mathematica, and it is barely an introduction to the subject.

This will get you part way to a solution

In[1]:= Clear[eqn, soln, u, x, y];
eqn = D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] == 0;
DSolve[{eqn}, u[x, y], {x, y}]

Out[1]= {{u[x, y] -> C[1][I x + y] + C[2][-I x + y]}}

So that says the generic solution to your problem consists of unknown functions C[1] and C[2] each of which accept a complex argument. Your u[x,0]==Sin[x] puts pretty tight requirements on what C[1] and C[2] are. Do you know just enough about complex functions to guess what C[1] and C[2] are? With that you might be able to fiddle a little and guess how to then modify those to satisfy your two remaining conditions.