Download presentation
Presentation is loading. Please wait.
1
The Wronskian and Linear Independence
MATH 374 Lecture 11 The Wronskian and Linear Independence
2
4.4: The Wronskian Using some ideas from linear algebra, we can find a way to check if certain sets of functions are linearly independent! For a review of the ideas we need, see this handout (Coddington – Introduction to ODE, pp ).
3
Check for Linear Independence
Suppose that each of the functions f1, f2, …, fn are at least (n-1) times differentiable on axb. If f1, f2, … , fn satisfy c1 f1 + c2 f2 + … + cn fn = 0 (1) for all x 2 [a, b], it follows that f1, f2, …, fn also must satisfy: 3
4
More Equations f1, f2, … , fn Satisfies
c1 f1’ + c2 f2’ + … + cn fn’ = 0 c1 f1’’ + c2 f2’’ + … + cn fn’’ = 0 (2) c1 f1(n-1) + c2 f2(n-1) + … + cn fn(n-1) = 0 for all x 2 [a, b]. 4
5
A System of Equations (for each x)
If we ask what choices of c1, c2, … , cn make (1) true, we are immediately led to a system of n equations in the n unknowns c1, c2, … , cn (for each x 2 [a, b]), because (2) must hold. 5
6
A System of Equations (for each x)
c1 f1(x) + c2 f2(x) + … + cn fn(x) = 0 c1 f1’(x) + c2 f2’(x) + … + cn fn’(x) = 0 (3) c1 f1(n-1)(x) + c2 f2(n-1)(x)+ … + cn fn(n-1)(x) = 0 for each x 2 [a, b]. 6
7
Matrix Form An equivalent form for (3) is the matrix equation: 7
8
Matrix Form 8
9
A Sufficient Condition for Linear Independence
If for some x0 2 [a, b], det[A(x0)] 0, it follows from Theorem 4 on the Coddington Handout (page 29) that system (3) with x = x0 has the unique solution c1 = c2 = … = cn = 0. Hence to have (1) hold for all x 2 [a, b], in particular for x = x0, we need ci = 0 for i = 1, 2, … , n. Therefore {f1, f2, … , fn } are linearly independent on [a, b] !! We’ve just proved a sufficient condition for linear independence: 9
10
A Sufficient Condition for Linear Independence
Theorem: Let f1, f2, … , fn be (n-1) times differentiable on axb. Define the Wronskian of {f1, f2, … , fn } to be the function: If W(x0) 0 for some x0 2 [a, b], then {f1, f2, … , fn } are linearly independent on [a, b]. 10
11
Notes W(x0) 0 for some x0 2 [a, b] is not a necessary condition for linear independence. There exist sets of functions {f1, f2, … , fn } with W(x) ´ 0 on [a, b] and {f1, f2, … , fn } are linearly dependent on [a, b] (see handout HW, problem 10). 11
12
Notes W(x0) 0 for some x0 2 [a, b] is equivalent to {f1, f2, … , fn } being linearly independent on [a, b], if we impose further conditions on {f1, f2, … , fn}. 12
13
A Necessary and Sufficient Condition for Linear Independence
Theorem 4.3: If on the interval axb, b0(x) 0, b0, b1, … , bn are continuous, and y1, y2, … , yn are solutions of the equation b0(x) y(n) + b1(x) y(n-1) + … + bn-1(x) y’ + bn(x) y = 0, then {y1, y2, … , yn} are linearly independent on [a,b] if and only if the Wronskian of {y1, y2, … , yn } differs from zero for at least one point x0 2 [a, b]. Proof: HW exercises for the n = 2 case. Induction for n > 2. 13
14
Example 1 For {1, x, x2}, Therefore these functions are linearly independent on any interval!
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.