I am attempting to solve a differential equation of the form:
$$ \frac{d^2 f}{d x^2} - \gamma^2 f = 0$$
I have set up and solved the characteristic equation as:
$$ \begin{align} z^2 - \gamma^2 z + 0z = 0 \\ z (z - \gamma^2 )= 0 \end{align}$$
which is satisfied when $z=0$ and when $z = \gamma^2$
There are two distinct roots ($r_1$ and $r_2$) hence the general solution should be of the form
$$ f(x) = A e^{r_1 x} + Be^{r_2 x}$$
In this case:
$$ \begin{align} f(x) &= A e^{0 \times x} + Be^{\gamma^2 x} \\ &= A + Be^{\gamma^2 x} \end{align} $$
Apparently, this solution is incorrect as in the answers it is given as:
$$ f(x) = Ae^{\gamma x} + B e^{-\gamma x}$$
I would like to know where I went wrong to give me the incorrect solution