# Transform the given differential equation or system into an equivalent system of first-order differential equations. x^{(3)}=(x')^{2}+\cos x

Transform the given differential equation or system into an equivalent system of first-order differential equations.
$$\displaystyle{x}^{{{\left({3}\right)}}}={\left({x}'\right)}^{{{2}}}+{\cos{{x}}}$$

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d2saint0
Consider the differential equation
$$\displaystyle{x}^{{{\left({3}\right)}}}={\left({x}'\right)}^{{{2}}}+{\cos{{x}}}$$
The objective is to transform tha given differential equetion into an equivalent system of first-order differeential equations. The third-order equation
$$\displaystyle{x}^{{{\left({3}\right)}}}={\left({x}'\right)}^{{{2}}}+{\cos{{x}}}$$
is equivalent to
$$\displaystyle{f{{\left({t},{x},{x}',{x}{''}\right)}}}={\left({x}'\right)}^{{{2}}}+{\cos{{2}}}$$
Now, let's be
$$\displaystyle{x}={x}_{{1}}$$
$$\displaystyle{x}'={x}_{{2}}={x}'_{{1}}$$
$$\displaystyle{x}{''}={x}_{{3}}={x}'{2}={x}{''}_{{1}}$$
The above substitution yields the system as written below
$$\displaystyle{x}'_{{1}}={x}_{{2}}$$
$$\displaystyle{x}'_{{2}}={x}_{{3}}$$
$$\displaystyle{x}'_{{3}}={x}^{{{2}}}_{2}+{{\cos{{x}}}_{{1}}}$$
are the three first order differential equations.