I am trying to derive
$y = \exp \left(\dfrac{a+b}2t \right) \left(k_1 \cosh \left(\dfrac{(a-b)t}2 \right) + k_2 \sinh \left(\dfrac{(a-b)t}2 \right) \right)$
From $y = c_1 \exp(at) + c_2 \exp(bt)$
I've managed to get the $ \exp \left ( \dfrac{a+b}2t \right)\cosh \left ( \dfrac{(a-b)t}{2} \right) $ part, but I cannot the hyperbolic sine part because of the minus sign. Remember, I am going from $y = c_1 \exp(at) + c_2 \exp(bt)$ to $y = \exp \left(\dfrac{a+b}2t \right) \left(k_1 \cosh \left(\dfrac{(a-b)t}2 \right) + k_2 \sinh \left(\dfrac{(a-b)t}2 \right) \right)$, not the other way around