Linear Independence and the Wronskian

Transcription

1 Linear Independence and the Wronskian MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018

2 Operator Notation Let functions p(t) and q(t) be continuous functions on the open interval I = (α, β). For any function φ(t) which is twice differentiable on I, we define the linear operator L as follows. L[φ] = φ + p φ + q φ

3 Operator Notation Let functions p(t) and q(t) be continuous functions on the open interval I = (α, β). For any function φ(t) which is twice differentiable on I, we define the linear operator L as follows. L[φ] = φ + p φ + q φ L[φ] is a function defined on I. We can evaluate this function as for all t I. L[φ](t) = φ (t) + p(t)φ (t) + q(t)φ(t)

4 Existence and Uniqueness of Solutions Theorem Consider the initial value problem L[y] = y + p(t)y + q(t)y = g(t) y(t 0 ) = y 0 y (t 0 ) = y 0 where p, q, and g are continuous on an open interval I that contains the point t 0. Then there is exactly one solution y = φ(t) to this problem and the solution is defined throughout the interval I.

5 Example Find the largest interval in which the solution to the initial value problem is certain to exist. (t 2 3t)y + t y (t + 3)y = 0 y(1) = 2 y (1) = 1

6 Principle of Superposition Theorem If y 1 and y 2 are two solutions to the differential equation L[y] = y + p(t)y + q(t)y = 0 then the linear combination y = c 1 y 1 + c 2 y 2 is also a solution for any values of the constants c 1 and c 2.

7 Solving the IVP Consider the IVP: L[y] = y + p(t)y + q(t)y = 0 y(t 0 ) = y 0 y (t 0 ) = y 0. Suppose y 1 (t) and y 2 (t) solve the ODE. Can a solution to the IVP be written as a linear combination of y 1 and y 2?

8 Solution Since y 1 and y 2 solve the ODE, then by the Principle of Superposition, y = c 1 y 1 + c 2 y 2 also solve the ODE. We need merely pick c 1 and c 2 so that the initial conditions are satisfied. c 1 y 1 (t 0 ) + c 2 y 2 (t 0 ) = y 0 c 1 y 1 (t 0) + c 2 y 2 (t 0) = y 0

9 Solution Since y 1 and y 2 solve the ODE, then by the Principle of Superposition, y = c 1 y 1 + c 2 y 2 also solve the ODE. We need merely pick c 1 and c 2 so that the initial conditions are satisfied. c 1 y 1 (t 0 ) + c 2 y 2 (t 0 ) = y 0 c 1 y 1 (t 0) + c 2 y 2 (t 0) = y 0 The solution to this linear system of equations can be found using Cramer s Rule.

10 Cramer s Rule c 1 y 1 (t 0 ) + c 2 y 2 (t 0 ) = y 0 c 1 y 1 (t 0) + c 2 y 2 (t 0) = y 0 Has solution c 1 = c 2 = y 0 y 2 (t 0) y 0 y 2(t 0 ) y 1 (t 0 )y 2 (t 0) y 1 (t 0)y 2 (t 0 ) = y 0 y 1 (t 0) + y 0 y 1(t 0 ) y 1 (t 0 )y 2 (t 0) y 1 (t 0)y 2 (t 0 ) = y 0 y 2 (t 0 ) y 0 y 2 (t 0) y 1(t 0 ) y 2 (t 0 ) y 1 (t 0) y 2 (t 0) y 1(t 0 ) y 0 y 1 (t 0) y 0 y 1(t 0 ) y 2 (t 0 ) y 1 (t 0) y 2 (t 0). Provided the denominator determinant is nonzero.

11 Wronskian Determinant The expression W (y 1, y 2 )(t 0 ) = y 1 (t 0 )y 2 (t 0) y 1 (t 0)y 2 (t 0 ) = y 1(t 0 ) y 2 (t 0 ) y 1 (t 0) y 2 (t 0) is called the Wronskian determinant.

12 Important Result Theorem Suppose that y 1 and y 2 are two solutions of and the initial conditions L[y] = y + p(t)y + q(t)y = 0 y(t 0 ) = y 0 y (t 0 ) = y 0 are assigned. Then it is always possible to choose constants c 1 and c 2 so that y = c 1 y 1 (t) + c 2 y 2 (t) satisfies the differential equation and initial conditions if and only if W (y 1, y 2 )(t 0 ) 0.

13 General Solution Theorem Suppose that y 1 and y 2 are two solutions of Then the family of solutions L[y] = y + p(t)y + q(t)y = 0. y = c 1 y 1 (t) + c 2 y 2 (t) with arbitrary coefficients c 1 and c 2 includes every solution to the ODE if and only if there is a point t 0 where the Wronskian of y 1 and y 2 is nonzero.

14 General Solution Theorem Suppose that y 1 and y 2 are two solutions of Then the family of solutions L[y] = y + p(t)y + q(t)y = 0. y = c 1 y 1 (t) + c 2 y 2 (t) with arbitrary coefficients c 1 and c 2 includes every solution to the ODE if and only if there is a point t 0 where the Wronskian of y 1 and y 2 is nonzero. Remarks: We call y = c 1 y 1 (t) + c 2 y 2 (t) with arbitrary coefficients the general solution. The set S = {y 1 (t), y 2 (t)} is called a fundamental set of solutions if and only if their Wronskian is nonzero.

15 Linear Independence Definition Two functions f and g are said to be linearly dependent on an interval I if there exist two constants k 1 and k 2, not both zero, such that k 1 f (t) + k 2 g(t) = 0 for all t I. If the equation above holds for all t I only if k 1 = k 2 = 0 then f and g are said to be linearly independent.

16 Examples Determine if the following pairs of functions are linearly dependent or independent on R. f (t) = e 3t and g(t) = e 3(t 1) f (t) = t 2 + t + 1 and g(t) = 3t 2 2t + 1

17 Wronskian The Wronskian can be used to determine if two functions are linearly independent on an interval.

18 Wronskian The Wronskian can be used to determine if two functions are linearly independent on an interval. Theorem If f and g are differentiable functions on an open interval I. and if W (f, g)(t 0 ) 0 for some t 0 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 I.

19 Wronskian The Wronskian can be used to determine if two functions are linearly independent on an interval. Theorem If f and g are differentiable functions on an open interval I. and if W (f, g)(t 0 ) 0 for some t 0 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 I. Proof. Suppose W (f, g)(t 0 ) 0 and suppose c 1 f (t) + c 2 g(t) is zero in I, then c 1 f (t 0 ) + c 2 g(t 0 ) = 0 c 1 f (t 0 ) + c 2 g (t 0 ) = 0

20 Examples Use the Wronskian to establish the linear dependence or independence of the following pairs of functions on R. f (t) = e 4t and g(t) = e 4t f (t) = 2t 3 and g(t) = 3t 3 f (t) = 1 and g(t) = cos t

21 Abel s Theorem (1 of 3) Theorem (Abel s Theorem) If y 1 and y 2 are solutions to the ODE L[y] = y + p(t)y + q(t)y = 0 where p and q are continuous on an open interval I, then the Wronskian W (y 1, y 2 )(t) is given by W (y 1, y 2 )(t) = ce p(t) dt where c is a constant that depends on y 1 and y 2, but not on t. Either W (y 1, y 2 )(t) = 0 for all t I (because c = 0) or W (y 1, y 2 )(t) 0 for all t I (because c 0).

22 Abel s Theorem (2 of 3) Proof. By assumption y q(t)y 1 = 0 y q(t)y 2 = 0. Multiplying the first equation by y 2 and multiplying the second equation by y 1 yields y 1 2 p(t)y 1 y 2 q(t)y 1 y 2 = 0 y 1 y 2 1y 2 + q(t)y 1y 2 = 0.

23 Abel s Theorem (3 of 3) Proof. y 1 2 p(t)y 1 y 2 q(t)y 1 y 2 = 0 y 1 y 2 1y 2 + q(t)y 1y 2 = 0 Adding the two equations produces y 1 y p(t) ( y 1 y 2 y 1 y 2) = 0 dw + p(t)w dt = 0

24 Examples Use Abel s Theorem to find the general form of the Wronskian for the following second order linear homogeneous ODEs. t 2 y + ty + (t 2 1)y = 0 (1 t 2 )y 2ty + 2y = 0

25 Consequences of Abel s Theorem The following theorem follows from Abel s Theorem: Theorem Let y 1 and y 2 be two solutions of L[y] = y + p(t)y + q(t)y = 0, where p and q are continuous on an open interval I. Then y 1 and y 2 are linearly dependent on I if and only if W (y 1, y 2 )(t) = 0 for all t I.

26 Homework Read Section 3.2 Exercises: 1 39 odd