Question

The general solution of the differential equation \[ \frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = e^x \cos 2x \] is:

• A. \( y(x) = c_1e^{2x} + c_2e^{-3x} - \frac{e^x}{20}(3\sin 2x - \cos 2x) \)
• B. \( y(x) = c_1e^{2x} + c_2e^{-3x} - \frac{e^x}{20}(3\sin 2x + \cos 2x) \)
• C. \( y(x) = c_1e^{2x} + c_2e^{-3x} + \frac{e^x}{8}(3\sin 2x - \cos 2x) \)
• D. \( y(x) = (c_1 + c_2x)e^{-3x} + \frac{e^x}{8}(3\sin 2x - \cos 2x) \)

Hint:

Use the method of undetermined coefficients to find the particular solution for equations with exponential and trigonometric forcing functions.

Answer 1

The Correct Option is B

We solve the given non-homogeneous differential equation by finding the complementary function and the particular integral. The complementary function comes from the homogeneous equation, and the particular solution is found using the method of undetermined coefficients.