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) =

Step-by-step 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.