therefore substitute this solution $in$ terms $ofuin$ the original differential equation.

$y_2^{\text{'}\text{'}}-16y_2=4$

$(u^{\text{'}\text{'}}{e}^{-4x}-8u^{\text{'}}{e}^{-4x}+16u{e}^{-4x})-16(u{e}^{-4x})=4$

$u^{\text{'}\text{'}}-()u^{\text{'}}=4e^{4x}$

This remains a second$-$order differential equation $inu.If,$ however, $we$ use the substitution $w=u^{\text{'}},we$ can reduce this $to$ a linear first$-$order differential equation $inw.$

$w^{\text{'}}-(|,)w=4e^{4x}$