Question

Solve the following differential equation:
\[ \frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y = 0 \]

• A. \( y = (C_1 + C_2 x) e^{-2x} \)
• B. \( y = C_1 e^{-2x} + C_2 x e^{-2x} \)
• C. \( y = C_1 e^{2x} + C_2 x e^{2x} \)
• D. \( y = (C_1 + C_2 x) e^x e^x \)

Hint:

When solving second-order linear differential equations with constant coefficients, if one of the roots is repeated, multiply the corresponding term by \( x \) to avoid duplication in the general solution.

Answer 1

The Correct Option is B

We are given the following differential equation: \[ \frac{d^2y}{dx^2} - 4y = 0 \] This is a linear second-order homogeneous differential equation with constant coefficients. 
Step 1: Write the characteristic equation The characteristic equation for this differential equation is: \[ r^2 - 4 = 0 \] \[ r^2 = 4 \quad \Rightarrow \quad r = \pm 2 \] 
Step 2: General solution For the roots \( r_1 = 2 \) and \( r_2 = -2 \), the general solution is: \[ y = C_1 e^{2x} + C_2 e^{-2x} \] Now, we need to apply the method of undetermined coefficients. Since the term \( e^{-2x} \) already appears in the solution, we multiply it by \( x \) to avoid duplication, which gives: \[ y = C_1 e^{-2x} + C_2 x e^{-2x} \] Thus, the solution is: \[ y = C_1 e^{-2x} + C_2 x e^{-2x} \]