Question

Solve (D2 - 3D + 2)y = x² + 2x + 1.​

Answer 1

Answer:

y = Aeˣ + Be²ˣ + (1/2)x² + (5/2)x + (15/4)

Step-by-step explanation:

First solve the homogeneous equation (D²-3D+2)y₋=0.

The characteristic equation is λ²-3λ+2 = (λ-1)(λ-2) = 0, which has roots 1 and 2.  So the general solution of the homogeneous equation is

  y₋ = Aeˣ + Be²ˣ

Next find a particular solution.  As there was no repeated root for the characteristic equation, look for a solution of the same form as the right hand side, so

 y₊ = ax² + bx + c

Then Dy₊ = 2ax + b  and  D²y₊ = 2a.

Substituting into the original equation gives

 2ax² + (2b-6a)x + (2c-3b+2a) = x² + x + 1.

Equating coefficients now gives

 a = 1/2,  b = 5/2,  c = 15/4

so a particular solution is

 y₊ = (1/2)x² + (5/2)x + (15/4).

The general solution of the given equation is then the general solution of the homogeneous equation plus the our particular solution, so

 y = y₋ + y₊

   = Aeˣ + Be²ˣ + (1/2)x² + (5/2)x + (15/4).

Hope that helps.