Question:

y” – y’ – 2y = 7e^2x

a) Use the method of undetermined coefficients to find the particular solution y_p of the second order non-homogeneous differential equation.

b) Find the general solution of the differential equation.

c) Find the differential equations given the initial conditions y(0) = 1/3 and y'(0) = -1/3

Answer:

Solving the auxiliary equation we have

y^2- y- 2y=0
(y+1)(y-2)=0
y=-1,2

y(x)= C1 e^(-x) +C2 e^(2x)

Since e^2x is a solution we will use A xe^(2x) for the method of undetermined coefficients.

y= Axe^(2x)
y’= 2Ax e^(2x) + Ae^(2x)
y”= 4Axe^(2x) +4A e^(2x)
So 4Axe^(2x) +4A e^(2x) -2Axe^(2x)-Ae^(2x)- 2Axe^(2x)= 7e^(2x)
3Ae^(2x)= 7e^(2x)
3A=7
A=7/3

So we have y(x)= C1 e^(-x) +C2e^(2x) +7/3 x e^(2x)

y(0)= 1/3
C1 +C2= 1/3

y’= -C1 e^(-x) + 2C2 e^(2x)+ 7/3 e^(2x) +14/3 x e^(2x)
y'(0)= =-C1 +2C2 +7/3= -1/3
3C2 +7/3= -1/3
3C2 =- 8/3
C2=- 8/9
C1= 3
y(x)= 3e^(-x) -8/9 e^(2x) + 7/3 x e^(2x)