Question

General solution of the differential equation (D² - 2D + 1) y = e^x is 

1. \(A{e^x} + B{e^{ - x}} + \frac{{{x^2}}}{2}{e^x}\)
2. \({e^x}\left( {A + Bx} \right) - \frac{{{x^2}}}{2}{e^x}\)
3. \(A{e^x} + B{e^{ - x}} - \frac{{{x^2}}}{2}{e^x}\)
4. \({e^x}\left( {A + Bx} \right) + \frac{{{x^2}}}{2}{e^x}\)

Answer 1

Correct Answer - Option 4 : \({e^x}\left( {A + Bx} \right) + \frac{{{x^2}}}{2}{e^x}\)

Concept:

General solution = C.F + P.I 

C.F = Complementary Function, P.I = Particular Integral

Calculation:

Given:

(D² - 2D + 1) y = e^x

For Particular Integral

\(y = \frac{{{e^x}}}{{{D^2} - 2D + 1}} = \frac{{x{e^x}}}{{2D - 2}} = \frac{{{x^2}{e^x}}}{2}\)

For Complementary Function

D² – 2D + 1 = 0

(D – 1)² = 0

D = 1, 1

C.F = (A + Bx) e^x

General solution = C.F + P.I 

\({e^x}\left( {A + Bx} \right) + \frac{{{x^2}}}{2}{e^x}\)