Question

Consider the differential equation 4y'' â 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 (ââ, â). The functions satisfy the differential equation and are linearly independent since W(ex/2, xex/2) = â  0 for ââ < x < â.

Answer 1

Check the Wronskian determinant:

$W(e^{x/2},xe^{x/2})=\begin{vmatrix}e^{x/2}&xe^{x/2}\\\frac12e^{x/2}&\left(1+\frac x2\right)e^{x/2}\end{vmatrix}=\left(1+\frac x2\right)e^x-\frac x2e^x=e^x\neq0$

The determinant is not zero, so the solutions are indeed linearly independent.