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Linear Independence and the Wronskian

- Two functions f and g are linearly dependent if

there exist constants c1 and c2, not both zero,

such that - for all t in I. Note that this reduces to

determining whether f and g are multiples of

each other. - If the only solution to this equation is c1 c2

0, then f and g are linearly independent. - For example, let f(x) sin2x and g(x) sinx

cosx, and consider the linear combination - This equation is satisfied if we choose c1 1,

c2 -2, and hence f and g are linearly

dependent.

Solutions of 2 x 2 Systems of Equations

- When solving
- for c1 and c2, it can be shown that
- Note that if a b 0, then the only solution to

this system of equations is c1 c2 0, provided

D ? 0.

Example 1 Linear Independence (1 of 2)

- Show that the following two functions are

linearly independent on any interval - Let c1 and c2 be scalars, and suppose
- for all t in an arbitrary interval (?, ? ).
- We want to show c1 c2 0. Since the equation

holds for all t in (?, ? ), choose t0 and t1 in

(?, ? ), where t0 ? t1. Then

Example 1 Linear Independence (2 of 2)

- The solution to our system of equations
- will be c1 c2 0, provided the determinant D

is nonzero - Then
- Since t0 ? t1, it follows that D ? 0, and

therefore f and g are linearly independent.

Theorem 3.3.1

- If f and g are differentiable functions on an

open interval I and if W(f, g)(t0) ? 0 for some

point t0 in I, then f and g are linearly

independent on I. Moreover, if f and g are

linearly dependent on I, then W(f, g)(t) 0 for

all t in I. - Proof (outline) Let c1 and c2 be scalars, and

suppose - for all t in I. In particular, when t t0 we

have - Since W(f, g)(t0) ? 0, it follows that c1 c2

0, and hence f and g are linearly independent.

Abels Theorem (Problem 4.2.36)

- Suppose y1 and y2 are solutions to the equation
- where p and q are continuous on some open

interval I. Then W(y1,y2)(t) is given by - where c is a constant that depends on y1 and y2

but not on t. - Note that W(y1,y2)(t) is either zero for all t in

I (if c 0) or else is never zero in I (if c ?

0).

Example 2 Wronskian and Abels Theorem

- Recall the following equation and two of its

solutions - The Wronskian of y1and y2 is
- Thus y1 and y2 are linearly independent on any

interval I, by Theorem. Now compare W with

Abels Theorem - Choosing c -2, we get the same W as above.

Theorem

- Suppose y1 and y2 are solutions to equation

below, whose coefficients p and q are continuous

on some open interval I - Then y1 and y2 are linearly dependent on I iff

W(y1, y2)(t) 0 for all t in I. Also, y1 and y2

are linearly independent on I iff W(y1, y2)(t) ?

0 for all t in I.

Summary

- Let y1 and y2 be solutions of
- where p and q are continuous on an open interval

I. - Then the following statements are equivalent
- The functions y1 and y2 form a fundamental set of

solutions on I. - The functions y1 and y2 are linearly independent

on I. - W(y1,y2)(t0) ? 0 for some t0 in I.
- W(y1,y2)(t) ? 0 for all t in I.

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