University:
Georgia Southern University
Course:
MATH 3230 | Ordinary Differential Equations
Academic year:

2024
Views:

386

Pages:

17
Author:

Jalen W.

Definition Let S = {y1 (x), y2 (x), . . . , yn (x)} be a set of n univariate functions and let D be a set on which all n functions are continuous. We say that S is linearly independent the only possible way that C1 y1 (x) + C2 y2 (x) + · · · + Cn yn (x) = 0 for all x in D is if Ci = 0 for all 1 ≤ i ≤ n. Example Show that S = {e αx , e βx } is linearly independent if and only if α 6= β. Example Show that S = {cos(x), sin(x)} is linearly independent. Example Show that S = {p(x), q(x)} is linearly independent if and only if p(x) 6= Cq(x) for some constant C . Definition The Wronskian of a set S = {y1 (x), y2 (x)}, denoted W (S), is given by W (S) = y1 (x) y2 (x) y10 (x) y20 (x) = y1 (x)y20 (x) − y2 (x)y10 (x). The Wronskian of a set S = {y1 (x), y2 (x), y3 (x)} is given by W (S) = y1 (x) y2 (x) y3 (x) y10 (x) y20 (x) y30 (x) y100 (x) y200 (x) y300 (x) This generalizes to a set S of n functions as well. Fact 2 A set S = {y1 (x), y2 (x), . . . , yn (x)} is linearly dependent if and only if W (S) = 0. Example Show that S = {sin(ax), cos(ax)} is linearly independent using the Wronskian. Definition An nth-order linear ODE of the form an (x) y (n) + an−1 (x) y (n−1) + an−2 (x) y (n−2) + · · · + a1 (x) y 0 + a0 (x) y = f (x) is said to be homogeneous if f (x) = 0 and nonhomogeneous if f (x) 6= 0. Fact 3: Principal of Superposition (Homogeneous Version) If y1 (x) and y2 (x) are two solutions to a homogeneous nth-order linear ODE, then y (x) = C1 y1 (x) + C2 y2 (x) is also a solution. Example Show that y1 (x) = sin(2x) and y2 (x) = cos(2x) are solutions to the ODE y 00 + 4y = 0. Now, show that y (x) = 2 sin(2x) + 3 cos(2x) is also a solution (i.e. verify the Principal of Superposition (Homogeneous Version) for this exmaple). Fact 4 The set of solutions to a homogeneous nth-order linear ODE is a vector space (over R) of dimension n, i.e. there is a basis of n solutions, S = {y1 (x), y2 (x), . . . , yn (x)}, that spans the entire vector space. That means that if y (x) is a solution to the nth-order linear ODE, then y (x) = C1 y1 (x) + C2 y2 (x) + · · · + Cn yn (x) for some constants C1 , C2 , . . . , Cn . Definition A fundamental set for a homogeneous nth-order linear ODE is a set of n linearly independent solutions to the ODE. Example Show that S = {sin(2x), cos(2x)} is a fundamental set for y 00 + 4y = 0. Definition Any solution to a nonhomogeneous nth-order linear ODE is called a particular solution. Example Show that y (x) = sin(2x) + x 2 is a particular solution to y 00 + 4y = 4x 2 + 2. Fact 5: Principal of Superposition (Nonhomogeneous Version) Suppose yh (x) is a solution to the homogeneous nth-order linear ODE an (x) y (n) + an−1 (x) y (n−1) + an−2 (x) y (n−2) + · · · + a1 (x) y 0 + a0 (x) y = 0 and yp (x) is a particular solution to the nonhomogeneous nth-order linear ODE an (x) y (n) + an−1 (x) y (n−1) + an−2 (x) y (n−2) + · · · + a1 (x) y 0 + a0 (x) y = f (x) then y (x) = Cyh (x) + yp (x) is still a solution to an (x) y (n) + an−1 (x) y (n−1) + an−2 (x) y (n−2) + · · · + a1 (x) y 0 + a0 (x) y for any real number C . = f (x) Example Recall that yh (x) = sin(2x) is a solution to y 00 + 4y = 0. It is easily shown that yp (x) = x 2 is a solution to y 00 + 4y = 4x 2 + 2. Verify that y (x) = 3 sin(2x) + x 2 is also a solution to y 00 + 4y = 4x 2 + 2 (i.e. verify the Principal of Superposition (Nonhomogeneous Version) works for this example.

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