y”+y’-2y=2t Please find general solution?

2 Answers

• General solution= Ke^(-2t)+Ke^t-t-1/2

• Note that the above answer is wrong as there should be two arbitrary constants.

  Find the complementary function by solving the auxiliary equation:

  y'' - y' - 2 = 0

  m² + m - 2 = 0

  (m + 2)(m - 1) = 0

  m + 2 = 0 OR m - 1 = 0

  m = -2 OR m = 1

  yᶜ = C₁℮ᵗ + C₂℮^(-2t)

  yᶜ = C₁℮ᵗ + C₂ / ℮^(2t)

  Find the particular integral by comparing coefficients:

  yᵖ = At + B

  yᵖ' = A

  yᵖ'' = 0

  yᵖ'' - yᵖ' - 2yᵖ = 2t

  A - 2(At + B) = 2t

  A - 2At - 2B = 2t

  -2At + (A - 2B) = 2t

  -2A = 2

  A = -1

  A - 2B = 0

  2B = A

  B = A / 2

  B = -½

  yᵖ = -t - ½

  Find the general solution by combining these two parts:

  y = yᶜ + yᵖ

  y = C₁℮ᵗ + C₂ / ℮^(2t) - t - ½