Problem

Given that

$$y = e^{3x}\cosh 2x$$

prove by induction that for $n \in \mathbb{N}$

$$\frac{d^n y}{dx^n} = e^{3x}\left(\frac{5^n+1}{2}\cosh 2x+\frac{5^n-1}{2}\sinh 2x\right)$$