Transform the given differential equation or system into an equivalent system of first-order differential equations.x ^ { \prime \prime } + 3 x' - 7x

shadsiei 2021-05-05 Answered

Transform the given differential equation or system into an equivalent system of first-order differential equations.
\(\displaystyle{x}''+{3}{x}'-{7}{x}={t}^{{{2}}}\)

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Expert Answer

StrycharzT
Answered 2021-05-06 Author has 22852 answers
The second-order equation
\(\displaystyle{x}{''}+{3}{x}'+{7}{x}={t}^{{{2}}}\)
is equivalent to equation
\(\displaystyle{f{{\left({t},{x},{x}'\right)}}}={t}^{{{2}}}-{7}{x}-{3}{x}'\)
Hence the substitutions \(\displaystyle{x}_{{{1}}}={x},{x}_{{{2}}}={x}'={x}_{{{1}}}\) yield the system
\(\displaystyle{x}'_{{{1}}}={x}_{{{2}}}\)
\(\displaystyle{x}'_{{{2}}}={t}^{{{2}}}_{\left\lbrace{1}\right\rbrace}-{7}{x}_{{{1}}}-{3}{x}_{{{2}}}\)
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