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\\eq0`

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