Do These Functions Form a Fundamental Set for the Differential Equation?

[KillerZ]

Homework Statement

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the solution.

$$4y^{''} - 4y^{'} + y = 0$$

$$e^{x/2}, xe^{x/2}$$

Homework Equations

$$Wr(y_1,...,y_n)=det\left(\begin{array}{ccc}y_1&\cdots&y_n\\y_1\prime&\cdots&y_n\prime\\\vdots&\vdots &\vdots\\y_1^{(n-1)}&\cdots&y_n^{(n-1)}\end{array}\right)$$

The Attempt at a Solution

$$e^{x/2}, xe^{x/2}$$

$$\frac{e^{x/2}}{2}, \frac{xe^{x/2}}{2} + e^{x/2}$$

$$Wr(e^{x/2}, xe^{x/2})= det\left(\begin{array}{ccc}e^{x/2}&xe^{x/2}\\\frac{e^{x/2}}{2}&\frac{xe^{x/2}}{2} + e^{x/2}\\\end{array}\right)$$

$$(e^{x/2})(\frac{xe^{x/2}}{2} + e^{x/2}) - (xe^{x/2})(\frac{e^{x/2}}{2})$$

$$\frac{xe^{x/2}}{2} + e^{x} - \frac{xe^{x/2}}{2} = e^{x} \neq 0$$

Therefore the functions are linearly independent and form a solution.$$y = c_{1}e^{x/2} + c_{2}xe^{x/2}$$

 

[LCKurtz]

I don't see a question. Your work looks correct as far as it goes. But have you shown that your two functions are actually solutions?