# Write the system of equations in the image in matrix form x'_1(t)=3x_1(t)-2x_2(t)+e^tx_3(t) x'_2(t)=sin(t)x_1(t)+cos(t)x_3(t) x'_3(t)=t^2x_1(t)+tx^2(t)+x_3(t)

Write the system of equations in the image in matrix form
${x}_{1}^{\prime }\left(t\right)=3{x}_{1}\left(t\right)-2{x}_{2}\left(t\right)+{e}^{t}{x}_{3}\left(t\right)$
${x}_{2}^{\prime }\left(t\right)=\mathrm{sin}\left(t\right){x}_{1}\left(t\right)+\mathrm{cos}\left(t\right){x}_{3}\left(t\right)$
${x}_{3}^{\prime }\left(t\right)={t}^{2}{x}_{1}\left(t\right)+t{x}^{2}\left(t\right)+{x}_{3}\left(t\right)$
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Step 1
Now , we need to write this system of equations in matrix form .
The given system of equations is ,
${x}_{1}^{\prime }\left(t\right)=3{x}_{1}\left(t\right)-2{x}_{2}\left(t\right)+{e}^{t}{x}_{3}\left(t\right)$
${x}_{2}^{\prime }\left(t\right)=\mathrm{sin}\left(t\right){x}_{1}\left(t\right)+\mathrm{cos}\left(t\right){x}_{3}\left(t\right)$
${x}_{3}^{\prime }\left(t\right)={t}^{2}{x}_{1}\left(t\right)+t{x}^{2}\left(t\right)+{x}_{3}\left(t\right)$
This system of equations is in 3 variables , ${x}_{1}\left(t\right),{x}_{2}\left(t\right),{x}_{3}\left(t\right)$
Therefore , the matrix formed will be a $3×3$ matrix .
Step 2
We can write down the system of equations in the form ${X}^{\prime }\left(t\right)=AX\left(t\right)$.
Hence , we get,
$\left[\begin{array}{c}{x}_{1}^{\prime }\left(t\right)\\ {x}_{2}^{\prime }\left(t\right)\\ {x}_{3}^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}3& -2& {e}^{t}\\ \mathrm{sin}\left(t\right)& 0& \mathrm{cos}\left(t\right)\\ {t}^{2}& t& 1\end{array}\right]\left[\begin{array}{c}{x}_{1}\left(t\right)\\ {x}_{2}\left(t\right)\\ {x}_{3}\left(t\right)\end{array}\right]$
Results: The system of equations can be written in matrix form as, $\left[\begin{array}{c}{x}_{1}^{\prime }\left(t\right)\\ {x}_{2}^{\prime }\left(t\right)\\ {x}_{3}^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}3& -2& {e}^{t}\\ \mathrm{sin}\left(t\right)& 0& \mathrm{cos}\left(t\right)\\ {t}^{2}& t& 1\end{array}\right]\left[\begin{array}{c}{x}_{1}\left(t\right)\\ {x}_{2}\left(t\right)\\ {x}_{3}\left(t\right)\end{array}\right]$

Jeffrey Jordon