We have found the general solution $y$ and its derivatives. Now we can substitute the given initial conditions.

Since $y(0)=0,$ we can state the following.

$y=c_1{c}_2{e}^{-8x}{c}_{3}x{e}^{-8x}$

$=c_1{c}_2$

Since $y^{\text{'}}(0)=1,$ we can state the following.

Since $y^{\text{'}\text{'}}(0)=-5,$ we can state the following.

$y^{\text{'}\text{'}}=64c_2{e}^{-8x}-16c_3{e}^{-8x}64c_{3}x{e}^{-8x}$

$-5=64c_2-()c_3$