Using Laplace transform methods, the solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is
A) $y = e ^ { t } + t e ^ { t } + t ^ { 2 } e ^ { t } / 2$
B) $y = e ^ { - t } + t e ^ { - t } + t ^ { 2 } e ^ { - t } / 2$
C) $y = e ^ { t } - t e ^ { t } + t ^ { 2 } e ^ { t } / 2$
D) $y = e ^ { - t } - t e ^ { - t } - t ^ { 2 } e ^ { - t } / 2$
E) $y = e ^ { - t } + t e ^ { - t } - t ^ { 2 } e ^ { - t } / 2$

Question

Quizplus · Accepted Answer

The correct solution involves transforming the differential equation using Laplace transforms, solving for Y(s) in the s-domain, and then finding the inverse Laplace transform to return to the time domain. The given differential equation, initial conditions, and the right-hand side that involves $e^{-t}$ suggest that the solution will involve terms that decay over time, hence the correct answer involves $e^{-t}$ and its derivatives, which is option B.

Using Laplace Transform Methods, the Solution Of y′′+2y′+y=e−t,y(0)=1,y′(0)=0y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0y′′+2y′+y=e−t,y(0)=1,y′(0)=0

Using Laplace Transform Methods, the Solution Of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$