1. ## PDE problem

I'm following an example for the separation of varibales method from a book and they went from
$Y_1 '' -pi^2 Y_1 = 0$
$Y_2 '' -4pi^2 Y_2 = 0$

to
$Y_1 (y) = Acosh(pi*y) + B_1 sinh(2pi*y)$
$Y_2 (y) = A_2 cosh(2pi*y) + B_2 sinh(2pi*y)$

Can someone explain how they made that jump?

Thanks

2. General solution to the differential equation $y''= k \cdot y$ is,

$y = A \cosh (\sqrt{k} x) + B \sinh (\sqrt{k} x)$

3. Thanks!