Question

The general solution of the differential equation $y'' + 4y' + 4y = 0$ is:

• A. $C_1 e^{2x} + C_2 e^{-2x}$
• B. $C_1 e^{-2x} + C_2 x e^{-2x}$
• C. $e^{-2x}(C_1 \cos 2x + C_2 \sin 2x)$
• D. $(C_1 + C_2 x)e^{-x}$

Hint:

Repeated roots → $(C_1 + C_2 x)e^{mx}$.

Answer 1

The Correct Option is B

Concept: Solve linear differential equation using auxiliary equation.

Step 1: Form auxiliary equation \[ m^2 + 4m + 4 = 0 \]

Step 2: Solve \[ (m + 2)^2 = 0 \Rightarrow m = -2, -2 \]

Step 3: Repeated roots case For repeated roots: \[ y = (C_1 + C_2 x)e^{mx} \]

Step 4: Substitute \[ y = (C_1 + C_2 x)e^{-2x} \] \[ \therefore \text{Correct answer is (B)} \]