Question

Solve the differential equation (D² - 2D - 3) y = 3e^2x

Answer 1

The given equation (D² – 2D – 3)y = 3e^2x … (1)

has auxiliary equation as: D² – 2D – 3 = 0

or D = 3, – 1 .…(2)

Thus the complementary function, C.F. = c₁e^3x + c2^e–x … (3)

For particular integral,

P.I = 1/D² - 2D - 3e^2x(comparable to  1/f(D) e^ax = e^ax/f(a), f(a) ≠ 0)

= 3/(2)² - 2(2) - 3 e^2x = -e^2x

Whence the complete solution, y = (c₁e^3x + c₂e^-x) - e^-2x