We wish to solve the PDE
$u_{tt} = 9 u_{xx} , \qquad -\infty < x < \infty , \quad t > 0$

Guess solutions of the form $u = e^{\alpha x+\beta t}$ (find a relationship between $\alpha$ and $\beta$).

Suppose you know $u(x,0) = e^{x}$. There are exactly two solutions of the above form, one with a positive $u_t(x,0)$ and one with negative $u_t(x,0)$. The first is:
$u = {}$
And the second is
$u = {}$

help (formulas)