Solve System of ODE's using Eulers method

Question

$Y'=AY$

$A=\begin{bmatrix}1&2&-1\\ \:1&0&1\\ \:4&-4&5\end{bmatrix}$

$Y(0)=\begin{pmatrix}-1\\ 0\\ 0\end{pmatrix}$

I have to solve this system using Eulers method, but what's Eulers method, like I know how to approach this using the eigenvalue method, but the problem explicitly states that it has to be solved using that method, yet my textbook doesn't provide an algorithm for such a method.

Answer 1

Euler method is a numerical procedure. For given time step $h>0$ one can calculate sequence of points that approximate the solution.

$y_{0}=Y(0)$

$y_{n+1}=y_{n}+h Ay_{n}$

So you have $ Y(nh)\approx y_{n}$. Note that this method does not give you continuous function and in general it is not accurate for large $n$ even for small $h$.