Question

A particular integral of the differential equation \[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = e^{2x} \sin x \] is

• A. \( \frac{e^{2x}}{10} (3 \cos x - 2 \sin x) \)
• B. \( -\frac{e^{2x}}{10} (3 \cos x - 2 \sin x) \)
• C. \( -\frac{e^{2x}}{5} (2 \cos x + \sin x) \)
• D. \( \frac{e^{2x}}{5} (2 \cos x - \sin x) \)

Hint:

For solving non-homogeneous differential equations, use the method of undetermined coefficients and guess a solution form based on the right-hand side.

Answer 1

The Correct Option is C

Step 1: Solve the homogeneous equation.
First, we solve the homogeneous part of the differential equation: \[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = 0. \] This is a second-order linear differential equation with constant coefficients. The solution is: \[ y_h = C_1 e^{2x} + C_2 e^{-x}. \]
Step 2: Solve the non-homogeneous equation.
For the non-homogeneous part, we use the method of undetermined coefficients. We guess a particular solution of the form: \[ y_p = A e^{2x} \cos x + B e^{2x} \sin x. \] Substitute this into the differential equation and solve for \( A \) and \( B \). After performing the calculations, we find that the particular solution is: \[ y_p = -\frac{e^{2x}}{5} (2 \cos x + \sin x). \]
Step 3: Final solution.
Thus, the correct answer is (C).