y8y16y2e4x yo1 y00Solutiony 8y 16y 0 m2 8m 16 0 m 4

y\'\'-8y\'+16y=2e^4x, y(o)=1, y\'(0)=0

Solution

y\'\' - 8y\' + 16y = 0

m^2 - 8m + 16 = 0

( m - 4 ) ( m - 4) = 0 ====> m = 4

y (x) = C1e^(4x) + C2 x e^(4x) =====> y(0)=1

1 = C1 + 0 ====> C1 = 1

y \' (x) =4e^(4x) + C2 x 4e^(4x) + C2e^(4x) =====> y \' (0) = 0

0 = 4*e^(4*0) + C2 *0 4e^(4*0) + C2e^(4*0)

0 = 4 + C2 ====> C2 = -4

y (x) = e^(4x) - 4x e^(4x)

to solve the particular solution:

Yp (x) = e^(4x)
Yp \' (x) = 4e^(4x)
Yp \'\' (x) = 16e^(4x)

16e^(4x) - 8 *4e^(4x) + 16*e^(4x) = 2e^4x

0 = 2e^4x ====> no particular

y (x) = e^(4x) - 4x e^(4x)

y[x] = e^(4 x) (1 - 4 x)