Solve by using the method of undetermined coefficients IVP y

Solve by using the method of undetermined coefficients IVP y\" + 4y + = (t + 3) e^-2t subject to y(0) y(0) = 5

Solution

First we solve homogeneous ode

y\'\'+4y\'+4y=0

Let, y=exp(kt) ,substituting gives

k^2+4k+4=0

k=-2

y=exp(-2t)(At+B)

Since, exp(-2t) and t exp(-2t) are already solutions to homogeneous ode so for particular solution we take teh guess

yp=exp(-2t)(Ct^2+Dt^3)

yp\'=-exp(-2t)t(-2C+2Ct-3Dt+2Dt^2)

yp\'\'=2 exp(-2t) (C-4Ct+3Dt+2Ct^2-6Dt^2+2Dt^3)

Substituting in ode gives

exp(-2t) (2C+6Dt)=(t+3)exp(-2t)

Comparing coefficients gives

2C=3,C=3/2

6D=1,D=1/6

Hence,

y=exp(-2t)(At+B)+exp(-2t)(3t^2/2+t^3/6)