Transform the differential equation

$\{\begin{array}{ll}{u}^{\u2034}(t)=\mathrm{sin}({u}^{\u2033}(t))-{u}^{2}(t),& t>0,\\ {u}^{(i)}={u}_{i}\phantom{\rule{1em}{0ex}}\text{for}i\in \{0,1,2\}\end{array}$

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 $\mathrm{sin}({u}^{\u2033}(t))$ term.

For far I tried setting $x:\equiv {u}^{\u2033}$ 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:

$\{\begin{array}{l}{y}^{\prime}(t)=x(t),\\ {x}^{\prime}(t)=\mathrm{sin}(x(t))-{u}^{2}(t),\\ {y}^{\u2033}(t)=\mathrm{sin}({y}^{\prime}(t))-{u}^{2}(t).\end{array}$

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.

$\{\begin{array}{ll}{u}^{\u2034}(t)=\mathrm{sin}({u}^{\u2033}(t))-{u}^{2}(t),& t>0,\\ {u}^{(i)}={u}_{i}\phantom{\rule{1em}{0ex}}\text{for}i\in \{0,1,2\}\end{array}$

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 $\mathrm{sin}({u}^{\u2033}(t))$ term.

For far I tried setting $x:\equiv {u}^{\u2033}$ 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:

$\{\begin{array}{l}{y}^{\prime}(t)=x(t),\\ {x}^{\prime}(t)=\mathrm{sin}(x(t))-{u}^{2}(t),\\ {y}^{\u2033}(t)=\mathrm{sin}({y}^{\prime}(t))-{u}^{2}(t).\end{array}$

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.