PDE4

$u_{xx}-u_t=0\,$

For this problem, guess that the solution is in the form $e^{ax+bt}\,$ . Plug the guess into the original PDE to find that:

$a^2-b=0\,$

$a = \pm \sqrt{b}\,$

Let $b = c 1$ , a constant. Adding together the two cases for $a$ , the homogeneous solution is

$u(x,t) = e^{ \sqrt{c_1}x + c_1t} + e^{ -\sqrt{c_1}x + c_1t}\,$