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 $c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - \frac{2}{3}x$.