TITLE: Problem regarding the Wronskian formula
QUESTION [6 upvotes]: I'm studying from G.Teschl ODE & Dynamical system and I got stucked at problem 3.20, which is:
Find a formula for the Wronskian $W(x,y)=\dot xy-\dot yx$ of the autonomous system $\ddot x + c_{1}\dot x + c_{0}x$. 
I know I can find the characteristic polynomial $\lambda^{2}+c_{1}\lambda+c_{0}=0$ where the can be calculated and that way try to find explicits solutions. But I don't think that's what is wanted where.
Thanks so much for your answers <3

REPLY [6 votes]: Let $x_1$ and $x_2$ be two fundamental solutions of given DE. Then
$x_1''+c_1x_1'+c_0x_1=0$
$x_2''+c_1x_2'+c_0x_2=0$
Multiply first by $x_2$ and second by $x_1$ and then on subtraction, you get
$x_1''x_2-x_2''x_1+c_1(x_1'x_2-x_2'x_1)=0$
$\implies W'=-c_1W$
On integration, $W=k.e^{-\int c_1dx}$, where $k$ is integration constant.