Transform the given differential equation or system into an equivalent system of first-order differential equations. x^{(3)}-2x''+x'=1+te^{t}

Ayaana Buck 2021-05-08 Answered
Transform the given differential equation or system into an equivalent system of first-order differential equations.
\(\displaystyle{x}^{{{\left({3}\right)}}}-{2}{x}{''}+{x}'={1}+{t}{e}^{{{t}}}\)

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Expert Answer

Khribechy
Answered 2021-05-09 Author has 4252 answers
Transform the given differential equation or system into an equivalent system of first-order differential equations.
\(\displaystyle{x}^{{{\left({3}\right)}}}-{2}{x}{''}+{x}'={1}+{t}{e}^{{{t}}}\) The third-order equation \(\displaystyle{x}^{{{\left({3}\right)}}}-{2}{x}{''}+{x}'={1}+{t}{e}^{{{t}}}\)
is equivalent to system
\(\displaystyle{f{{\left({t},{x},{x}',{x}{''},{x}{''}\right)}}}={1}+{t}{e}^{{{t}}}-{x}'+{2}{x}{''}\)
Hense the substitutions \(\displaystyle{x}_{{1}}={x},{x}_{{2}}={x}'={x}'_{{1}},{x}_{{3}}={x}{''}={x}'_{{2}}\)
yield the system
\(\displaystyle{x}'_{{1}}={x}_{{2}}\)
\(\displaystyle{x}'_{{2}}={x}_{{3}}\)
\(\displaystyle{x}'_{{3}}={1}+{t}{e}^{{{t}}}-{x}_{{2}}+{2}{x}_{{3}}\) is a system of first-order equation.
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