I'm asked to decide if I should solve the system

$\dot{y}=\left(\begin{array}{cc}-600& 400\\ 400& -600\end{array}\right)y,\phantom{\rule{1em}{0ex}}t\in [{t}_{0},{t}_{e}],\phantom{\rule{1em}{0ex}}y({t}_{0})={y}_{0}$

with either the explicit Euler method or the implicit Euler method.

Using the explicit Euler method I would get the updating scheme

${y}_{n+1}=\left(\begin{array}{cc}1-600h& 400h\\ 400h& 1-600h\end{array}\right){y}_{n}$

where the eigenvalues of the driving matrix is

${\lambda}_{1}=401-600h,$

${\lambda}_{2}=-399-600h.$

For the solution to be stable these need to be less than one which gives the conditions

$h\ge \frac{4}{6}=\frac{2}{3},$

$h\ge -\frac{4}{6}=-\frac{2}{3}.$

The last condition doesn't say anything but the first condition seems restrictive since I can't choose $h$ as small as possible.

If I instead were to use the implicit Euler method I would get the updating scheme

${y}_{n+1}={\left(\begin{array}{cc}1-600h& 400h\\ 400h& 1-600h\end{array}\right)}^{-1}{y}_{n}.$

Now I can't solve for the eigenvalues of this system but I've heard the implicit Euler is unconditionally stable so it shouldn't matter.

So is the answer that I should choose implicit Euler because it is unconditionally stable or am I missing something? The order of consistency of both is 1 so that should not matter.

$\dot{y}=\left(\begin{array}{cc}-600& 400\\ 400& -600\end{array}\right)y,\phantom{\rule{1em}{0ex}}t\in [{t}_{0},{t}_{e}],\phantom{\rule{1em}{0ex}}y({t}_{0})={y}_{0}$

with either the explicit Euler method or the implicit Euler method.

Using the explicit Euler method I would get the updating scheme

${y}_{n+1}=\left(\begin{array}{cc}1-600h& 400h\\ 400h& 1-600h\end{array}\right){y}_{n}$

where the eigenvalues of the driving matrix is

${\lambda}_{1}=401-600h,$

${\lambda}_{2}=-399-600h.$

For the solution to be stable these need to be less than one which gives the conditions

$h\ge \frac{4}{6}=\frac{2}{3},$

$h\ge -\frac{4}{6}=-\frac{2}{3}.$

The last condition doesn't say anything but the first condition seems restrictive since I can't choose $h$ as small as possible.

If I instead were to use the implicit Euler method I would get the updating scheme

${y}_{n+1}={\left(\begin{array}{cc}1-600h& 400h\\ 400h& 1-600h\end{array}\right)}^{-1}{y}_{n}.$

Now I can't solve for the eigenvalues of this system but I've heard the implicit Euler is unconditionally stable so it shouldn't matter.

So is the answer that I should choose implicit Euler because it is unconditionally stable or am I missing something? The order of consistency of both is 1 so that should not matter.