Abel's formula, confusion!

Hello everyone,

The differiental equation i solved was:

dW/dx = -p(x)W.

I solved a diffferential equation above and it equaled,

W(x) = ce^(-Intgral[ p(x) dx ]), where c is a constant.


And i guess its a famous formual and its name is Abel's formula.
But now they want me to explain,

Why does Abel's formula imply that the Wronskian W(y1,y2) is either zero everywhere or nonzero everywhere?

Well I know the following:

Suppose that y1 and y2 are two solutions of the homoegenous second order linear equation,


y'' + p(x)y' + q(x)y = 0; //equation 1

on an open interaval I on which p and q are continuus.

(a) If y1 and y2 are linearly dependent, then W(y1,y2) = 0 on I.
(b) If y1 and y2 are linearly independent, then W9y1,y2) != 0 at each point on I.

Thus, given two solutions of Eq. 1, there are just two possiblities: the Wronksian W is identically zero if the solutions are linearly dependent; The wronskian is never zero if the solutions are linearly independent.

But this doens't explain why Abel's fomrula implys that the Wronkian W(y1,y2) is either zero everywhere or nonzero everywhere?



Thanks for listening,
:thumb: