Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 4 y = \cos ( 2 x )$ is
A) $y _ { p } = A \cos ( 2 x )$
B) $y _ { p } = A \cos ( 2 x ) + B \sin ( 2 x )$
C) $y _ { p } = A x \cos ( 2 x )$
D) $y _ { p } = A x \cos ( 2 x ) + B \sin ( 2 x )$
E) $y _ { p } = A x \cos ( 2 x ) + B x \sin ( 2 x )$

Question

Quizplus · Accepted Answer

The homogeneous solution includes terms like $\cos(2x)$ and $\sin(2x)$, so the particular solution must be multiplied by $x$ to avoid duplication, leading to $y_p = Ax\cos(2x) + Bx\sin(2x)$.

Without Solving for the Undetermined Coefficients, the Correct Form of a Particular