Second order nonhomogeneous differential equations?

Question

How do you find the general solution of this equation? y'' + 2y' + y = 2e^t. The answer is y = (c1)e^(-t) + (c2)te^(-t) + t^2*e^(-t). Thanks.

Answer 1

First solve the homogeneous equation:
y" + 2y' + y = 0
The characteristic (or auxiliary) equation is r^2 + 2r + 1 = 0 <=> (r+1)^2 = 0, repeated roots at r = -1.
So the complementary function will be
y_c = Ae^(-t) + Bx e^(-t) = e^(-t) (A + Bt).

For the particular solution, we try y_p = ke^t, so we get
y"_p + 2y'_p + y_p = ke^t + 2ke^t + ke^t = 4ke^t = 2e^t, so we need 4k = 2, i.e. k = 1/2.

So the general solution is y = y_c + y_p
= (A + Bt) e^(-t) + (1/2) e^t.

Answer 2

As typical, the final answer is the sum of a "specific" or specific answer Y(x) + the final answer of the the homogeneous equation, y" + y' - 2y = 0 . the superb area is the sum of a polynomial and an exponential. the main suitable thank you to discover a definite answer is to discover a definite answer Y1(x) for case a million: suitable area =2 x^2, and a definite answer Y2(x) for case 2: suitable area = 3 e^(-x). Then Y(x) = Y1(x) + Y2(x) is a answer for the certainly case: suitable area = 2x^2 + 3 e^(-x) . Case a million: y" + y' - 2y = 2x^2 it particularly is lifelike to look for a answer Y1(x) = y(x) = a 2d degree polynomial : y(x) = - x^2 + ax + b so - 2y = 2x^2 -2ax - 2b. the ingredient is that the 2x^2 term suits the 2x^2 on the superb area. y' = -2x +a ; ... y" = - 2 Left area y" + y' - 2y = (-2) + (-2x +a) + ( 2x^2 - 2ax - 2b) = suitable area 2x^2 2x^2 - (2a + 2) x + (a - 2b - 2) = 2x^2 2a + 2 = 0 ; ... a - 2b - 2 = 0 a = -- a million; ... b = -- (3/2) Y1 = y = -- (x^2 + x + 3/2) ------------ Case 2: y" + y' - 2y = 3 e^(-x) attempt Y2 = y = Ae^(-x) y" + y' - 2y = Ae^(-x) - Ae^(-x) - 2Ae^(-x) = 3e^(-x) A = -- 3/2 ----------------- Y = Y1 + Y2 = -- ( x^2 + x + 3/2 + (3/2) e^(-x) ) ------ Homogeneous eq: y" + y' - 2y = 0 recommendations e^(rx), with r^2 + r -- 2 = 0; roots r = a million, -2 usual answer Ae^x + B e^(-2x) ---- usual answer of inhomogeneous eq: y = Y(x) + Ae^x + Be^(-2x) preliminary situations: y(0) = 0 and y'(0) = a million. those situations supply 2 linear equations which verify A and B.