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.

\(\displaystyle{x}^{{{\left({3}\right)}}}={\left({x}'\right)}^{{{2}}}+{\cos{{x}}}\)

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.