Matrix-vector representation for a system of ODE's

Question

I am aiming to explicitly write the matrix-vector representation of this system: $$\begin{aligned}y'_1 &= 5y_2 - y_1 + y_3\\ y_2' &= 3y_1 - y_2 + t^2\\ y_3' &= y_3 - ty_2\end{aligned}$$

This is what I have so far: $\left[\begin{matrix}y_1'\\ y_2'\\y_3'\end{matrix}\right]=\left[\begin{matrix}-1 & \;\;5 &\; 1\\ \;\;\;3 & -1 & \;0 \\ \;\;\;0 & \;\;? &\; 1 \end{matrix}\right]\cdot \left[\begin{matrix}0\\t^2\\?\end{matrix}\right]$

Just not sure how to attack $-ty_2.$
Any help would be appreciated, thanks guys.

Answer 1

Just set $$ \left(\begin{array}{c} y'_1\\ y'_2\\ y'_3 \end{array} \right)= \left(\begin{array}{ccc} -1&5&1\\ 3&-1&0\\ 0&-t&1 \end{array} \right) \left(\begin{array}{c} y_1\\ y_2\\ y_3 \end{array} \right)+ \left(\begin{array}{c} 0\\ t^2\\ 0 \end{array} \right). $$

Answer 2

You have the following system of odes:

$\begin{align*} y_1'&= -y_1+5y_2+y_3\\ y_2'&=3y_1-y_2+t^2\\ y_3'&=y_3-ty_2 \end{align*}$

Then make $u=\begin{bmatrix} y_1&y_2&y_3 \end{bmatrix}^T$ and you get

$$u'= \begin{bmatrix}-1&5&1\\ 3&-1&0\\ 0&-t&1 \end{bmatrix}u+ \begin{bmatrix}0\\t^2\\0 \end{bmatrix}$$