Let $y_1(x)$ and $y_2(x)$ be solution of a second order ODE. If $y_1(x)$ and $y_2(x)$ are linearly independent on an interval $I$, then they are linearly independent on any interval containing I.
I need to prove if the above statement is true or give a counter-example if it is false.
I think it should be true. But being terrible at proofs, I don't know how to write a good proof for this.
I thought of using the Wronskian argument but a query came up. I know that if $y_1(x)$ and $y_2(x)$ are linearly independent then the Wronskian is non-zero at some point. But is the converse true?
This statement is very trivial. If no linear combination $c_1y_1(x)+c_2y_2(x)$ is the zero function on $I$, then that stays true on any interval containing $I$.
On the other hand, if $y(x)=c_1y_1(x)+c_2y_2(x)$ is the zero function on $I$, then $y(x_0)=y'(x_0)=0$ for any inner point of $I$. This then means that it has to be the zero solution everywhere where the differential equation exists.