Question

If the function $y=e^{4x}+2e^{-x}$ is a solution of the differential equation $\frac{\frac{d^{3}y}{d{x}^3}-13\frac{dy}{dx}}{y}=k$, then the value of $\frac{k}{3}$ is

• A. -4
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

$y=e^{4x}+2e^{-x}$
$\frac{dy}{dx}=4e^{4x}-2e^{-x}\Rightarrow \frac{d^{2}y}{d{x}^2}=16e^{4x}+2e^{-x}$
$\frac{d^{3}y}{d{x}^3}=64e^{4x}-2e^{-x}$
$\therefore \frac{\frac{d^{3}y}{d{x}^3}-13\frac{dy}{dx}}{y}=\frac{\left(64e^{4x}-2e^{-x}\right)-13\left(4e^{4x}-2e^{-x}\right)}{e^{4x}+2e^{-x}}=\frac{12e^{4x}+24e^{-x}}{e^{4x}+2e^{-x}}=12$
$\therefore \frac{k}{3}=4$