If, show that


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


It is given that, $y=500 e^{7 x}+600 e^{-7 x}$


$\frac{d y}{d x}=500 \cdot \frac{d}{d x}\left(e^{7 x}\right)+600 \cdot \frac{d}{d x}\left(e^{-7 x}\right)$

$=500 \cdot e^{7 x} \cdot \frac{d}{d x}(7 x)+600 \cdot e^{-7 x} \cdot \frac{d}{d x}(-7 x)$

$=3500 e^{7 x}-4200 e^{-7 x}$

$\therefore \frac{d^{2} y}{d x^{2}}=3500 \cdot \frac{d}{d x}\left(e^{7 x}\right)-4200 \cdot \frac{d}{d x}\left(e^{-7 x}\right)$

$=3500 \cdot e^{7 x} \cdot \frac{d}{d x}(7 x)-4200 \cdot e^{-7 x} \cdot \frac{d}{d x}(-7 x)$

$=7 \times 3500 \cdot e^{7 x}+7 \times 4200 \cdot e^{-7 x}$

$=49 \times 500 e^{7 x}+49 \times 600 e^{-7 x}$

$=49\left(500 e^{7 x}+600 e^{-7 x}\right)$

$=49 y$

Hence, proved.

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