Question

Use the substitution v=y' to write each second-order equation as a system of two first-order differential equations (planar system). y^{\prime \prime}

Differential equations
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asked 2021-06-16

Use the substitution \(y'=v\) to write each second-order equation as a system of two first-order differential equations (planar system). \(\displaystyle{y}^{''}+\mu{\left({t}^{{{2}}}-{1}\right)}{y}^{''}+{y}={0}\)

Answers (1)

2021-06-17

\(\displaystyle{y}''+\mu{\left({t}^{{{2}}}-{1}\right)}{y}''+{y}={0}\)
\(\displaystyle{y}{''}+\mu{\left({t}^{{{2}}}-{1}\right)}{y}'+{y}={0}\)
We use the substitution
\(y'=v\)
\(\displaystyle\Rightarrow{y}{''}={v}'\)
We substitute y' and y'' in equation
\(\displaystyle{y}{''}=-\mu{\left({t}^{{{2}}}-{1}\right)}{y}'-{y}\)
we get \(\displaystyle{v}'=-\mu{\left({t}^{{{2}}}-{1}\right)}{v}-{y}\)
Therefore, we get system of first-order equations
\(y'=v\)
\(v'=-\mu(t^{2}-1)v-y\)

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