Question

If $y=500e^{7x}+600e^{-7x},$ then the value of $\frac{d^{2}y}{d{x}^2}-49y$ is

• A. $7(500e^{7x}-600e^{-7x})$
• B. $49(500e^{7x}+600e^{-7x})$
• C. $500e^{7x}+600e^{-7x}$
• D. $0$

Answer 1

The correct option is D $0$

Given $y=500e^{7x}+600e^{-7x}$
Differentiating with respect to $x$ we get,
$y_1=\frac{dy}{dx}=500\left(e^{7x}\right)\left(7\right)+600\left(e^{-7x}\right)(-7)=7(500e^{7x}-600e^{-7x})$
Differentiating again with respect to $x$ we get,
$y_2=\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=7\left[500e^{7x}\right(7)-600e^{-7x}(-7\left)\right]=49(500e^{7x}+600e^{-7x})=49y$
$\therefore y_2-49y=0$