The problem instructions read like so:
Show that the given functions are linearly independent on the interval I and ﬁnd
a second-order linear homogeneous equation having the pair as a fundamental set of
I'm able to prove that they are linearly independent easily by simply using the Wronskian Matrix to get
However, I don't know how to find the equation based off of what I'm given.
I know the answer to the question (it's in the book) so I will post it for reference of what I'm looking for.
please show me the method to find this equation based off of the fundamental set given!
Ok, I think I know what to do now, lol. I was on the right track... Just define and , basically divide the first two equations (from my failed system) by the coefficient of y'' to get:
From the first:
Now substitute into the second:
That should help...
Remember we defined .