Question

solve ( D² + 4D + 4 )y = $e^{2x} - e^{-2x}$

Answer 1

$\bold{ANSWER≈}$

Since y=e2x is part of the homogeneous solution (from solving the characteristic equation r2−4r+4=(r−2)2=0 ), we can use reduction of order.

Let y=e2xz . Differentiating yields Dy=e2x(z′+2z) and D2y=e2x(z′′+4z′+4z) . Then, substituting this into the original differential equation yields

e2x(z′′+4z′+4z)−4⋅e2x(z′+2z)+4⋅e2xz=x3e2x,

which simplifies to

z′′=x3.

Integrating twice in succession yields

z=C1x+C2+120x5.

Rewriting this in terms of y then gives us

y=(C1x+C2)e2x+120x5e2x.

Hence, a particular solution/integral is y=120x5e2x.

Answer 2

Answer:

Let y=e2xz . Differentiating yields Dy=e2x(z′+2z) and D2y=e2x(z′′+4z′+4z) . Then, substituting this into the original differential equation yields

e2x(z′′+4z′+4z)−4⋅e2x(z′+2z)+4⋅e2xz=x3e2x,

which simplifies to

z′′=x3.

Integrating twice in succession yields

z=C1x+C2+120x5.

Rewriting this in terms of y then gives us

y=(C1x+C2)e2x+120x5e2x.

Hence, a particular solution/integral is y=120x5e2x.

$thanks$