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Differential Equations

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Moderate

Nonhomogeneous Linear System: Undetermined Coefficients

DIFFEQ-@FHWS3

Consider $x'=Ax+\left(\begin{array}{c}t^2\\\ 2t\end{array}\right)$, where $ A=\left(\begin{array}{cc} 3&-4\\\ 1&-1\end{array}\right).$

The general solution can be written as $x=x_h+x_p$ where $x_h$ is a general solution of $x'=Ax$ and $x_p$ is a particular solution. One can find a particular solution by variation of parameters or by undermined coefficients. In the latter method, we simply guess a form $x_p= a t^2+b t+c$ where $a,b,c$ are three vectors.

The three vectors are given by

A

$a=\left(\begin{array}{c}1\\\ 1\end{array}\right),\ b=\left(\begin{array}{c}-2\\\ -2\end{array}\right),\ c=\left(\begin{array}{c}-6\\\ -4\end{array}\right).$

B

$a=\left(\begin{array}{c}1\\\ 0\end{array}\right),\ b=\left(\begin{array}{c}0\\\ 2\end{array}\right),\ c=\left(\begin{array}{c}0\\\ 0\end{array}\right).$

C

$a=\left(\begin{array}{c}-1\\\ -1\end{array}\right),\ b=\left(\begin{array}{c}-14\\\ -10\end{array}\right),\ c=\left(\begin{array}{c}-26\\\ -16\end{array}\right).$

D

No vectors $a,b,c$ can make $x=at^2+bt+c$ a particular solution.