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?