Question

Transform the given differential equation or system into an equivalent system of first-order differential equations. x ^ { \prime \prime } + 4 x - x ^

Matrices
ANSWERED
asked 2021-06-16

Transform the given differential equation or system into an equivalent system of first-order differential equations.

Answers (1)

2021-06-17
The second-order equation
\(\displaystyle{x}{''}+{4}{x}'-{x}^{{{3}}}={0}\)
is equivalent to equation
\(\displaystyle{f{{\left({x},{x}'\right)}}}={x}^{{{3}}}-{4}{x}\)
Hence the substitutions \(\displaystyle{x}_{{{1}}}={x},{x}_{{{2}}}={x}'={x}_{{{1}}}\) yield the system
\(\displaystyle{x}'_{{{1}}}={x}_{{{2}}}\)
\(\displaystyle{x}'_{{{2}}}={x}^{{{3}}}_{\left\lbrace{1}\right\rbrace}-{4}_{{{1}}}\)
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