Variation of Parameters

Order of Reduction Consider the nonhomogeneous nth order linear ODE with constant coefficients below. an y (n) + an−1 y (n−1 ) + · · · + a1 y 0 + a0 y = f (x) Suppose that S = {y1 (x), y2 (x), . . . , yn (x)} is a fundamental set for the homogeneous version of the ODE, an y (n) + an−1 y (n−1) + · · · + a1 y 0 + a0 y = 0. Then, a particular solution to the nonhomogeneous ODE must cannot be a linear combination of the functions in S, i.e. yp (x) = u1 (x)y1 (x) + u2 (x)y2 (x) + · · · + un (x)yn (x) for some functions u1 (x), u2 (x), . . . , un (x). Matrices Let A, A1 , A2 , . . . , An  y1 (x)  y10 (x)  00  A =  y1 (x)  ..  . (n−1) y1 be n × n matrices defined by: y2 (x) y20 (x) y200 (x) .. . (n−1) (x) y2 y3 (x) y30 (x) y300 (x) .. . (n−1) (x) y3 ... ... ... .. . yn (x) yn0 (x) yn00 (x) .. . (n−1) (x) . . . yn     ,   (x) and Ai is exactly the same as A but with the ith column replaced by (0, 0, · · · , 0, f (x))T . Now, let Wi = |Ai | and recall that W (S) = |A|. Variation of Parameters For an nth order nonhomogeneous linear ODE, with S = {y1 (x), y2 (x), . . . , yn (x)} being a fundamental set for the homogeneous version of the ODE, a particular solution to the ODE exists of the form, yp (x) = u1 (x)y1 (x) + u2 (x)y2 (x) + · · · + un (x)yn (x), where Z ui (x) = for 1 ≤ i ≤ n. Wi dx an (x)W (S) Special Note for this Chapter In this Chapter, we are only dealing with 2nd order ODEs, so we just have W (S) = y1 (x) y10 (x) y2 (x) y20 (x) , W1 = 0 f (x) y2 (x) y20 (x) , W2 = y1 (x) y10 (x) 0) f (x) , and, thus, Z u1 (x) = −f (x)y2 (x) dx, u2 (x) = y1 (x)y20 (x) − y10 (x)y2 (x) Z f (x)y1 (x) dx. y1 (x)y20 (x) − y10 (x)y2 (x) Example Find the general solution to the ODE. y 00 − y 0 = e x . Example Find the general solution to the ODE. y 00 − y 0 − 2y = e x . Example Find the general solution to the ODE. y 00 − y 0 − 2y = x 2 . Example Find the general solution to the ODE. y 00 + 4y 0 + 13y = 4e −2x sin(3x) Example Show that S = {x −3 , x 2 } are solutions to the homogeneous linear ODE with NONconstant coefficients x 2 y 00 + 2xy 0 − 6y = 0. Now, use Variation of Parameters to find the general solution to the nonhomogeneous ODE x 2 y 00 + 2xy 0 − 6y = x 2 .