Question

Consider the differential equation4y'' − 4y' + y = 0; ex/2, xex/2.Verify that the functions ex/2 and xex/2 form a fundamental set of solutions of the differential equation on the interval (−[infinity], [infinity]).The functions satisfy the differential equation and are linearly independent since W(ex/2, xex/2) =

Answer 1

W = (1/4)(x + 1)e^(2x) - (1/4)xe^(2x)

= (1/4)(x + 1 - x)e^(2x)

W = (1/4)e^(2x)

Explanation:

Answer 2

Explanation:

Let y1 and y2 be (e^x)/2, and (xe^x)/2 respectively.

The Wronskian of them functions be

W = (y1y2' - y1'y2)

y1 = (e^x)/2 = y1'

y2 = (xe^x)/2

y2' = (1/2)(x + 1)e^x

W = (1/4)(x + 1)e^(2x) - (1/4)xe^(2x)

= (1/4)(x + 1 - x)e^(2x)

W = (1/4)e^(2x)

Since the Wronskian ≠ 0, we conclude that functions are linearly independent, and hence, form a set of fundamental solutions.