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

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

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
$$y'=v$$
$$v'=-\mu(t^{2}-1)v-y$$