 # Transform the differential equation { <mtable columnalign="left left" rowspacing=".2 daktielti 2022-07-07 Answered
Transform the differential equation

into an equivalent system of first order ordinary differential equations.
I know how to carry out this procedure for linear systems but I am stuck because of the $\mathrm{sin}\left({u}^{″}\left(t\right)\right)$ term.
For far I tried setting $x:\equiv {u}^{″}$ and $y:\equiv {u}^{\prime }$ and then substituting into the equation to obtain multiple differential equations in x and y but don't know how to simplify them to obtain the desired result:
$\left\{\begin{array}{l}{y}^{\prime }\left(t\right)=x\left(t\right),\\ {x}^{\prime }\left(t\right)=\mathrm{sin}\left(x\left(t\right)\right)-{u}^{2}\left(t\right),\\ {y}^{″}\left(t\right)=\mathrm{sin}\left({y}^{\prime }\left(t\right)\right)-{u}^{2}\left(t\right).\end{array}$
For context: This is the first part of a two part question where in the second part one is asked to find the iterative rule for Eulers method.
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Given the equation
$\begin{array}{}\text{(1)}& {u}^{‴}\left(t\right)=\mathrm{sin}\left({u}^{″}\left(t\right)\right)-\left(u\left(t\right){\right)}^{2},\end{array}$
we set
$\begin{array}{}\text{(2)}& v\left(t\right)={u}^{\prime }\left(t\right),\end{array}$
and
$\begin{array}{}\text{(3)}& w\left(t\right)={v}^{\prime }\left(t\right)={u}^{″}\left(t\right);\end{array}$
so that
$\begin{array}{}\text{(4)}& {w}^{\prime }\left(t\right)={v}^{″}\left(t\right)={u}^{‴}\left(t\right);\end{array}$
then the equation (1) may be written
$\begin{array}{}\text{(5)}& {w}^{\prime }\left(t\right)=\mathrm{sin}\left(w\left(t\right)\right)-\left(u\left(t\right){\right)}^{2},\end{array}$
which together with
$\begin{array}{}\text{(6)}& {u}^{\prime }\left(t\right)=v\left(t\right),\end{array}$
$\begin{array}{}\text{(7)}& {v}^{\prime }\left(t\right)=w\left(t\right),\end{array}$
becomes a first-order system in the three variables u, v, w.
We may in fact "unravel" (5)-(7) to recover (1), simply by substituting (3) and (4) back into (5).
Thus, equation (1) is equivalent to the system (5)-(7).

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