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)

Step 1
Now , we need to write this system of equations in matrix form .
The given system of equations is ,
${\displaystyle x_1^{\prime }\left(t\right)=3x_1\left(t\right)-2x_2\left(t\right)+e^t{x}_3\left(t\right)}$
${\displaystyle x_2^{\prime }\left(t\right)=\sin \left(t\right)x_1\left(t\right)+\cos \left(t\right)x_3\left(t\right)}$
${\displaystyle x_3^{\prime }\left(t\right)=t^2{x}_1\left(t\right)+t{x}^2\left(t\right)+x_3\left(t\right)}$
This system of equations is in 3 variables , ${\displaystyle x_1\left(t\right),x_2\left(t\right),x_3\left(t\right)}$
Therefore , the matrix formed will be a ${\displaystyle 3\times 3}$ matrix .
Step 2
We can write down the system of equations in the form $X^{\prime }(t)=AX(t)$.
Hence , we get,
${\displaystyle \left[\begin{array}{c}{x}_1^{\prime }\left(t\right)\\ x_2^{\prime }\left(t\right)\\ x_3^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}3& -2& e^t\\ \sin \left(t\right)& 0& \cos \left(t\right)\\ t^2& t& 1\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\end{array}\right]}$
Results: The system of equations can be written in matrix form as, ${\displaystyle \left[\begin{array}{c}{x}_1^{\prime }\left(t\right)\\ x_2^{\prime }\left(t\right)\\ x_3^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}3& -2& e^t\\ \sin \left(t\right)& 0& \cos \left(t\right)\\ t^2& t& 1\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\end{array}\right]}$