Question

The general solution of the differential equation $\frac{d^{2}y}{d{x}^2}+8\left(\frac{dy}{dx}\right)+16y=0$ is

• A. $(A+Bx)e^{5x}$
• B. $(A+Bx)e^{-4x}$
• C. $(A+B{x}^2)e^{4x}$
• D. $(A+B{x}^4)e^{4x}$

Answer 1

The correct option is B

$(A+Bx)e^{-4x}$

Find the general solution of the differential equation

Given, $\frac{d^{2}y}{d{x}^2}+8\left(\frac{dy}{dx}\right)+16y=0$

Put $D=\frac{d}{dx}$

$$\begin{gathered} \Rightarrow (D^2+8D+16)y=0 \\ \Rightarrow D^2+8D+16=0 \\ \Rightarrow D^2+4D+4D+16=0 \\ \Rightarrow D(D+4)+4(D+4)=0 \\ \Rightarrow (D+4)(D+4)=0 \\ \Rightarrow D=-4,-4 \end{gathered}$$

Since, roots are real and equal.

$\therefore$General Solution is $y=(A+Bx)e^{-4x}$

Hence, option (B) is the correct answer.