Question

The general solution of the differential equation \[ \frac{d^2 y}{dx^2} + \frac{dy}{dx} + y = 2e^{3x} \] is given by

• A. \( y = (c_1 + c_2x) e^x + \frac{e^{-3x}}{8} \)
• B. \( y = (c_1 + c_2x) e^x + \frac{e^{3x}}{8} \)
• C. \( y = (c_1 + c_2x) e^x + \frac{e^{3x}}{8} \)
• D. \( y = (c_1 + c_2x) e^x + \frac{e^{-3x}}{8} \)

Hint:

Solve second-order linear differential equations by finding the homogeneous and particular solutions separately.

Answer 1

The Correct Option is C

Step 1: Solving the second-order linear differential equation.
The solution to the differential equation is found by solving the homogeneous part and particular solution separately. The final solution is \( y = (c_1 + c_2x) e^x + \frac{e^{3x}}{8} \).

Step 2: Conclusion.
Thus, the correct answer is option (C).

Final Answer: \[ \boxed{\text{(C) } y = (c_1 + c_2x) e^x + \frac{e^{3x}}{8}} \]