Find the second order derivatives of the function.

Let $y=e^{x} \sin 5 x$

$\frac{d y}{d x}=\frac{d}{d x}\left(e^{x} \sin 5 x\right)=\sin 5 x \cdot \frac{d}{d x}\left(e^{x}\right)+e^{x} \frac{d}{d x}(\sin 5 x)$

$=\sin 5 x \cdot e^{x}+e^{x} \cdot \cos 5 x \cdot \frac{d}{d x}(5 x)=e^{x} \sin 5 x+e^{x} \cos 5 x \cdot 5$

$=e^{x}(\sin 5 x+5 \cos 5 x)$

$\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left[e^{x}(\sin 5 x+5 \cos 5 x)\right]$

Then,

$=(\sin 5 x+5 \cos 5 x) \cdot \frac{d}{d x}\left(e^{x}\right)+e^{x} \cdot \frac{d}{d x}(\sin 5 x+5 \cos 5 x)$

$=(\sin 5 x+5 \cos 5 x) e^{x}+e^{x}\left[\cos 5 x \cdot \frac{d}{d x}(5 x)+5(-\sin 5 x) \cdot \frac{d}{d x}(5 x)\right]$

$=e^{x}(\sin 5 x+5 \cos 5 x)+e^{x}(5 \cos 5 x-25 \sin 5 x)$

$=e^{x}(10 \cos 5 x-24 \sin 5 x)=2 e^{x}(5 \cos 5 x-12 \sin 5 x)$