The correct form of the particular solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - x }$ is
A) $y _ { p } = A e ^ { - x }$
B) $y _ { p } = A x e ^ { - x }$
C) $y _ { p } = A x ^ { 2 } e ^ { - x }$
D) $y _ { p } = A x ^ { 3 } e ^ { - x }$
E) none of the above

Question

Quizplus · Accepted Answer

The correct form of the particular solution involves increasing the power of $x$ in the trial solution until it is linearly independent from the solutions of the homogeneous equation. The homogeneous equation $y'' + 2y' + y = 0$ has solutions of the form $e^{-x}$, so to avoid duplication and ensure linear independence, the particular solution must include $x^2$, making $y_p = A x^2 e^{-x}$ the correct choice.

The Correct Form of the Particular Solution Of y′′+2y′+y=e−xy ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - x }y′′+2y′+y=e−x

The Correct Form of the Particular Solution Of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - x }$