Question

The solution of the differential equation \[ \frac{d^2y}{dx^2} + 3y = -2x \] is

• A. \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - \frac{2}{3}x \)
• B. \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - \frac{4}{5}x \)
• C. \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - 2x^2 + \frac{4}{3} \)
• D. \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - 2x^2 + \frac{9}{4} \)

Hint:

For non-homogeneous differential equations, find the complementary solution for the homogeneous part and the particular solution for the non-homogeneous part.

Answer 1

The Correct Option is A

The given differential equation is a non-homogeneous second-order linear equation.
The complementary solution is \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x \), and the particular solution is of the form \( -\frac{2}{3}x \).
Therefore, the total solution is \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - \frac{2}{3}x \).