Find the solution of the following Differential Equation

Question

(D^2 + 6 D + 9) y = 17e^2x

a. y = c1 e^-3x + c2 x e^-3x + 17/25 (e^2x)

b. y = c1 e^-3x + c2 x e^-3x + 17/9 (e^2x)

c. y = c1 e^2x + c2 x e^2x + 17/25 (e^-3x)

d. y = c1 e^2x + c2 x e^2x + 17/9 (e^-3x)

Answer 1

1) Find the complementary solution by solving first the characteristic equation D^2+6D+9 (assuming the D is the differential operator) to find r1,r2= -3, -3 (repeated roots).

2) With repeated real roots the complementary solution will be yc=c1e^rx+c2te^rx

3) Since g(x) = 17e^2x the particular solution will have the form Ae^2x

4) Substiture Ae^2x into the original problem to find D^2y=4Ae^2x, Dy=2Ae^2x, y=Ae^2x

5) Match coefficients to find A=17/25, thus yc=25e^2x

6) y=yc+yp

7) answer is a)