If $y=500 e^{7 x}+600 e^{-7 x},$ show that $\frac{d^{2} y}{d x^{2}}=49 y$.

Given that,$y=500 e^{7 x}+600 e^{-7 x}$.
Differentiating with respect to $x$:
$\frac{d y}{d x} = 500 \cdot \frac{d}{d x}(e^{7 x}) + 600 \cdot \frac{d}{d x}(e^{-7 x})$
$= 500 \cdot e^{7 x} \cdot 7 + 600 \cdot e^{-7 x} \cdot (-7)$
$= 3500 e^{7 x} - 4200 e^{-7 x}$.
Differentiating again with respect to $x$:
$\frac{d^{2} y}{d x^{2}} = 3500 \cdot \frac{d}{d x}(e^{7 x}) - 4200 \cdot \frac{d}{d x}(e^{-7 x})$
$= 3500 \cdot e^{7 x} \cdot 7 - 4200 \cdot e^{-7 x} \cdot (-7)$
$= 24500 e^{7 x} + 29400 e^{-7 x}$.
Factoring out $49$:
$= 49(500 e^{7 x} + 600 e^{-7 x})$.
Since $y = 500 e^{7 x} + 600 e^{-7 x}$,we have:
$\frac{d^{2} y}{d x^{2}} = 49 y$.
Hence,proved.