Is there an explicit ODE solution for this system?

Find the solution to the second order ODE with constant coefficients?

• Find the solution to the second order ODE with constant coefficients. First solve for the homogeneous solution, and then solve the non-homogeneous case by the method of undetermined coefficients. State the general and particular form for the given set of initial conditions. Show that the homogeneous solutions form a fundamental set. y"+2y'+y=2e^(-t), y(0)=-3

• Answer:

  Find the complementary function by solving the auxiliary equation: y'' + 2y' + y = 0 m² + 2m + 1 = 0 (m + 1)² = 0 m + 1 = 0 m = -1 yᶜ = (C₁t + C₂)℮^(-t) Find the particular integral by comparing coefficients: yᵖ = At²℮^(-t) yᵖ' = (2At - At²)℮^(-t) yᵖ'' = (At² - 4At + 2A)℮^(-t) yᵖ'' + 2yᵖ' + yᵖ = 2℮^(-t) (At² - 4At + 2A)℮^(-t) + 2(2At - At²)℮^(-t) + At²℮^(-t) = 2℮^(-t) (At² - 4At + 2A)℮^(-t) + (4At - 2At²)℮^(-t) + At²℮^(-t) = 2℮^(-t) 2A℮^(-t) = 2℮^(-t) 2A = 2 A = 1 yᵖ = t²℮^(-t) Find the general solution by combining these two parts: y = yᶜ + yᵖ y = (C₁t + C₂)℮^(-t) + t²℮^(-t) y = (t² + C₁t + C₂)℮^(-t) Find the particular solution by solving for the constants: When t = 0, y = 2 C₂ = 2 y' = [(C₁ - C₂) - (C₁ - 2)t - t²]℮^(-t) When t = 0, y' = -3 C₁ - C₂ = -3 C₁ = C₂ - 3 C₁ = 2 - 3 C₁ = -1 y = (t² - t + 2)℮^(-t) y = (t² - t + 2) / ℮ᵗ