Task:
Consider the system of first-order ODES: \begin{align*} y_1' =&5y_2-y_1+y_3\\ y_2'=&3y_1-y_2+t^2\\ y_3'=&y_3-ty_2 \end{align*}
Write out the matrix-vector representation of this system
$y(t+h)\approx Fy(t)+g$
Clearly define the elements of $y'$, $y$, $F$ and $g$.
Question:
How do I approach this? I know that I'm meant to rearrange it somehow. And how would I account for t's in the system?
I would post my working out, but I actually have no clue where to start. I was taught that the elements of F and g may be functions of t as well as h. So I'm guessing accounting for t's in the system would have something to do with g...
Hint:
For $h\ll 1$, $y'(t)\approx\frac{y(t+h)-y(t)}{h}$. Now, solve for $y(t+h)$ and insert the ODEs for $y'(t)$.
Edit: Complete solution
We first write the ODEs in terms of matrices, with $y(t)=(y_1(t),y_2(t),y_3(t))^T$ and $y'(t)=(y_1'(t),y_2'(t),y_3'(t))^T$: \begin{equation*} y'(t)=\left(\begin{matrix}-1 & 5 & 1\\3 & -1 & 0\\0&-t&1\end{matrix}\right)y(t)+\left(\begin{matrix}0\\t^2\\0\end{matrix}\right) \end{equation*} Then we realize that we should give the answer in terms of $y(t+h)$, indicating that we should use the formula \begin{equation*} \frac{y(t+h)-y(t)}{h}\approx y'(t) \end{equation*} We solve this formula for $y(t+h)$, and insert $y'(t)$ from above: \begin{align*} y(t+h)&\approx y(t)+h y'(t)\\ &=\left(\begin{matrix}1 & 0 & 0\\ 0 & 1 & 0\\0&0&1\end{matrix}\right)y(t)+h\left(\begin{matrix}-1 & 5 & 1\\3 & -1 & 0\\0 & -t & 1\end{matrix}\right)y(t)+h\left(\begin{matrix}0\\t^2\\0\end{matrix}\right) \end{align*} Now, we bring these equations into the required form: \begin{align*} y(t+h)&\approx \underbrace{\left(\begin{matrix}1-h & 5h & h\\3h & 1-h & 0\\0 & -th & 1+h\end{matrix}\right)}_{F(t,h)}y(t)+\underbrace{\left(\begin{matrix}0\\ht^2\\0\end{matrix}\right)}_{g(t,h)} \end{align*}