Understanding the Wronskian Function in Mathematics
Explore the concept of the Wronskian function, its significance in differential equations, and how it determines linear dependence or independence of solutions. Examples illustrate the application of the Wronskian in determining the relationship between different functions.
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The Wronskian function: linear dependence and independence Dr. Aruna Kulkarni
In mathematics, the Wronskian is a determinant introduced by Jozef Hoene- Wronskian. It is used in the study of differential equations. It shows linear independence in a set of solutions.
Consider the equation L(y)=0 Where L(y) = ? + ?1? + ?2? = 0 If 1??? 2are two solutions of above equation then Wronskian W( 1, 2) is defined as
? 1,2 = 12 1 2 = 1 2 1 2 It is a function , and its value at x is denoted by ?( 1, 2)(?).
Theorem Two solutions 1, 2,?? L(y) = 0 are linearly independent on an interval I if, and only if, ? 1, 2 ? 0 for all x in I.
The functions 1,2defined in following examples exist for < ? < . Determine whether they are linearly dependent or independent there. [I] 1? = ?, 2(?) = ?2? Solution: we have 1? = ?, 2(?) = ?2? then 1(x)= 1 and 2(x)= 2?2? W( 1, 2)= 1 2 ? ?2? 1 2?2? = 2??2? ?2?= ?2?(2? 1) 0 L.I. 1 2 =
[II] 1? = cos?, 2(x)=sin? Solution: from above we get 1? = ????, 2(x)=cos? W( 1, 2)= 1 2 = cos? sin? sin? cos?= cos2? +sin2? = 1 value of determinant is not zero so they are linearly independent. 1 2
[III] 1? = ?2, 2(x)=5?2 solution: from above we have 1= 2x, 2= 10x ?( 1, 2)= 1 2 1 2 = 10?3 10?3 = 0 so they are linearly dependent. =?25?2 2? 10?
[IV] (?) = ??? 1? = ????, 2 Solution: from above we have 1? = ????, ?( 1, 2)= sin? ??? = ???????? ???? = ???? + ????? ????? ???? = ????? + ???? ????? ???? = ?2???2? ???2x = -???2? ???2x = (???2? + ???2x) = -1 0 ?? are linearly independent. 2? = ???? cos? ????
Assignment 1? = ????, 2(x)= 3(???+ ? ??). Thank You.
