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 $K$ is :

• A. 4
• B. 6
• C. 9
• D. 12

Answer 1

Answer: (D)

$y=e^{4x}+2e^{-x};y_1=4e^{4x}-2e^{-x};y_2=16e^{4x}+2e^{-x};y_3=64e^{4x}-2e^{-x}$
Now, $y_3-13y_1=\left(64e^{4x}-2e^{-x}\right)-13\left(4e^{4x}-2e^{-x}\right)=12e^{4x}+24e^{-x}$
$=12\left(e^{4x}+2e^{-x}\right)=12y$
$\therefore \frac{y_3-13y_1}{y}=12\Rightarrow D$