# Why can't this simple second-order differential equation be solved?

I'm trying to solve the following partial differential equation: $$f(x,y) = \partial_x^2f(x,y)-\partial_y^2f(x,y)$$ for which one obvious solution would be: $$f(x,y)=e^{2x+\sqrt{3}y}$$ An even more general solution would be: $$f(x,y)=c_1e^{\cosh(\alpha)\,x+\sinh(\alpha)\,y}+c_2e^{-(\cosh(\alpha)\,x+\sinh(\alpha)\,y)}$$ for arbitrary $$\alpha,\,c_1$$ and $$c_2$$.

When trying to let Mathematica solve this equation via

DSolve[f[x, y] == D[f[x, y], {x, 2}] - D[f[x, y], {y, 2}], f[x, y], {x, y}]


it just returns the input. Did I set up DSolve the wrong way?

• Every soft has its limitations. BTW, where is the PDE under consideration used? – user64494 Oct 21 '20 at 17:18

## 1 Answer

Unfortunately, DSolve is unable to solve many simple PDEs. Here, it is successful only with substantial human assistance. Assume f[x, y] -> fx[x] fy[y]. Then,

sx = DSolveValue[1 == fx''[x]/fx[x] - fy''[y]/fy[y], fx[x], x] /. {C -> c1x, C -> c2x}
(* c1x E^(-((x Sqrt[fy[y] + (fy''[y]])/Sqrt[fy[y]])) +
c2x E^((x Sqrt[fy[y] + (fy''[y]])/Sqrt[fy[y]]) *)


The quantity

sx[[2, 2, 2]]/x
(* Sqrt[fy[y] + (fy''[y]]/Sqrt[fy[y]] *)


must be a constant, so

sy = DSolveValue[%^2 == c^2, fy[y], y] /. {C -> c1y, C -> c2y}
(* c2y E^(-Sqrt[-1 + c^2] y) + c1y E^(Sqrt[-1 + c^2] y) *)


The solution of the PDE then is

sxy = sx sy /. sx[[2, 2, 2]] -> c x
(* (c1x E^(-c x) + c2x E^(c x)) (c2y E^(-Sqrt[-1 + c^2] y) + c1y E^(Sqrt[-1 + c^2] y)) *)


where the five constants may be complex numbers. The two solutions given in the question are special cases. The accuracy of this result can be verified by

Simplify[f[x, y] == D[f[x, y], {x, 2}] - D[f[x, y], {y, 2}] /.
f -> Function[{x, y}, Evaluate[sxy]]]
(* True *)


Probably, this simple PDE could be solved by hand more rapidly than with Mathematica.