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

nitraiddQ 2021-05-01 Answered
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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Expert Answer

d2saint0
Answered 2021-05-02 Author has 29136 answers
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.
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