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Differential Equations

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Moderate

Nonhomogeneous Linear System: Laplace Transform

DIFFEQ-3EFSSK

Consider the system of equations:

$$ x'=\left(\begin{array}{cc} 2&-5\\\ 1&-2\end{array}\right)x+ \left(\begin{array}{c} 0\\\ \sin t\end{array}\right) $$

...with initial condition $x(0)=\left(\begin{array}{c}1\\\ -1\end{array}\right)$.

The Laplace transform of $x(t)$, or $X=\mathcal{L}(x(\cdot))$ is given by

A

$X(s)=\frac{1}{(s^2+1)^2}\left(\begin{array}{c} s^3+7s^2+s+2\\\ -s^3+3s^2+1\end{array}\right)$

B

$X(s)=\frac{1}{(s^2+1)^2}\left(\begin{array}{c} -5\\\ s-2\end{array}\right)$

C

$X(s)=\frac{1}{s^2+1}\left(\begin{array}{c} s^3+7s^2+s+2\\\ -s^3+3s^2+1\end{array}\right)$

D

$X(s)=\frac{1}{s^2+1}\left(\begin{array}{c} -5\\\ s-2\end{array}\right)$