# DSolve fails to solve second order PDE

I have a very simple example of a second order PDE:

eqn = D[u[x, y], x, y] + u[x, y] == 0;
sol = DSolve[eqn, u, {x, y}]


and Mathematica cannot solve it! What is wrong!?

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In the tutorial on second-order PDE's it says "DSolve can find the general solution for a restricted type of homogeneous linear second-order PDEs..." Unfortunately this equation is not of that type (it has a non-vanishing non-principal part). So I don't think there's anything wrong as such, it's just that Mathematica can't solve this PDE. –  Simon Woods Mar 26 '13 at 20:48

When I run it Mathematica solves it perfectly well, but symbolically:

Clear[x, y, u] (* Just in case you have any of this predefined thereby confusing things *)

eqn = D[u[x, y], x, y] + u[x, y] == 0;
sol = DSolve[eqn, u, {x, y}]


Now this may not give you what you expected or wanted. So take a closer looks at your original code, to understand what may have gone wrong.

First -- Did you intend the Mathematica partial derivative function (D[]) to differentiate successively with respect to x and y? Alternatively you may have wanted something more like:

D[f,{x,n}]


which gives the multiple derivative.

Second -- Where you add u[x, y] to D[u[x, y], x, y], recognize that the x and y in u[x, y] (on the right of your + sign) are not the x and y in D[u[x, y], x, y]. D[] encapsulates the scope of the x and y. It doesn't reach beyond itself.

Three -- You use u[x, y] as a function both within D and in what you add to it, but you haven't either defined it prior or made it a pure function. As I don't know what you need to accomplish, I can't really comment further on this.

I hope this helps. Good luck.

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OPs equation looks like a 2D harmonic oscillator. Shouldnt the result simply be a two dimensional exponential? E.g $u(x,y)=exp(I(x+y))$ would be (one) solution to the above equation. –  A friendly helper Mar 26 '13 at 14:51
Nice to have a @Afriendlyhelper around... Welcome to the site. –  Jagra Mar 26 '13 at 15:27
@Afriendlyhelper is correct, it's the non homogeneous 2D wave equation in a different coordinate system (make the change $\xi=x+y,\ \eta=x-y$ to get it).Now the PDE is linear so you can combine the homogeneous solution with a special solution of the non homogeneous one. A special case for it is the $\exp(x+y)$ so the general solution is $F(x)+G(y)+\exp(x+y)$. –  Spawn1701D Mar 26 '13 at 20:48