# Consider the IVP <mtable columnalign="right left right left right left right left right left

Consider the IVP
$\begin{array}{r}{y}^{″}=-y\end{array}$
for $t\ge 0$, and $y\left(0\right)=1$, ${y}^{\prime }\left(0\right)=2$.

I have rewritten this differential equation as a system of first-order ODE's such that
$\begin{array}{r}{u}^{\prime }=v\\ {v}^{\prime }=-u\end{array}$
with $u\left(0\right)=1,v\left(0\right)=2$.

The solution is $y=2\mathrm{sin}\left(t\right)+\mathrm{cos}\left(t\right)$, ${y}^{\prime }=2\mathrm{cos}\left(t\right)-\mathrm{sin}\left(t\right)$

I am asked to perform one step of Euler's method with $h=0.5$, and determine if Euler's method is stable for this problem.

For the first part, I find that one step of Euler's method yields
$\begin{array}{r}{y}_{1}={y}_{0}+hf\left({t}_{0},{y}_{0}\right)=1+\left(0.5\right)\left(2\mathrm{cos}\left(0\right)-\mathrm{sin}\left(0\right)\right)=2.\end{array}$
But how do I determine if Euler's method is stable? I know that for the equation ${y}^{\prime }=\lambda y$, Euler's method is stable for $|1+h\lambda |\le 1$, but since this problem is in a different form I'm not sure what I need to do here.
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Bornejecbo
Applying the Euler iteration procedure we have
$\left\{\begin{array}{l}{u}_{k}={u}_{k-1}+h{v}_{k-1}\\ {v}_{k}={v}_{k-1}-h{u}_{k-1}\end{array}$
or
$\left(\begin{array}{c}{u}_{k}\\ {v}_{k}\end{array}\right)=\left(\begin{array}{cc}1& h\\ -h& 1\end{array}\right)\left(\begin{array}{c}{u}_{k-1}\\ {v}_{k-1}\end{array}\right)$
or
${U}_{k}={M}^{k}{U}_{0}$
this sequence converges as long as the eigenvalues of $M$ have absolute value less than 1. Here the $M$ eigenvalues are $1±ih$ with absolute value $\sqrt{1+{h}^{2}}>1$ so the Euler procedure diverges.
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Layla Velazquez
You apply the Euler step to the first-order system.
${u}_{1}={u}_{0}+h{u}_{0}^{\prime }={u}_{0}+h{v}_{0},\phantom{\rule{0ex}{0ex}}{v}_{1}={v}_{0}+h{v}_{0}^{\prime }={v}_{0}-h{u}_{0}.$
As the Euler step is tangential to the convex solution curve, it will always move outwards, away from the center of the concentric exact solution curves.