Wronskian to prove linear independence

[process91]

Homework Statement

Let v_1,v_2 be any two solutions of the differential equation y''+ay'+by=0 such that \frac {v_2}{v_1} is not constant, and let f(x) be any solution of the differential equation as well.

Use the properties of the Wronskian to prove that constants c_1,c_2 exist such that:

c_1 v_1(0) + c_2 v_2(0) = f(0), \qquad c_1 v_1 '(0) + c_1 v_1 '(0) = f' (0)

Homework Equations

Here are the relevant properties of the Wronskian, defined as W(x)=v_1(x) v_2 '(x) - v_2(x)v_1 '(x):

Let W be the Wronskian of two solutions v_1, v_2 of the differential equation y'' + ay' +by =0.
All the following holds:
W' +aW =0
W(x) = W(0)e^{-ax}
W(0) = 0 \iff \frac{v_2}{v_1} \text{is constant}

The Attempt at a Solution

\frac{v_2}{v_1} is not constant, so W(0) \ne 0, and therefore for some constant d we have
dW(0)=f(0)
d(v_1(0) v_2 '(0) - v_2(0)v_1 '(0)) = f(0)
[dv_2'(0)]v_1(0) + [-dv_1'(0)]v_2(0) = f(0)

So for our solution, c_1 = dv_2'(0) and c_2 = -dv_1'(0), but this leads to

[dv_2'(0)]v_1'(0) + [-dv_1'(0)]v_2'(0) = f'(0)=0

Which is not always true.

 

[process91]

Doh! - nevermind. Is there a way to delete a post?