\(\displaystyle{y}''+\mu{\left({t}^{{{2}}}-{1}\right)}{y}''+{y}={0}\)

\(\displaystyle{y}{''}+\mu{\left({t}^{{{2}}}-{1}\right)}{y}'+{y}={0}\)

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

\(y'=v\)

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