[EDIT]: Your Wronskian is correct, just not simplified as much as it could be. If you simplify it a bit further, I think your result will pop right out.
The problem :
Show that the functions , and are linearly independent if
Solution :
I calculated the wronskian, and my final answer is:
Clearly,
The problem here is with
Since , the following numbers can't be zero:
Now, I stopped!
- b-a
- a-c
- c-b
How I can prove that the expression can't be zero so that wronskian will be non zero and hence the function are linearly independent?
Just plug in your primed variables. Try things. Note that
The simplification is, for me, about 5 lines' worth. If you're taking differential equations, you should definitely be prepared to do that much algebra.
This forum is about helping people get unstuck. It is not about giving people answers. So plug things in, show your work, and let me know if you get stuck.
Consider the symmetry of the original expression. a, b, and c are all treated the same, right? You could probably replace a with b, b with c, and c with a without changing the expression (or maybe you'd get a minus sign - that's not important). But (a-c)(a-b) is not that symmetric. How could you make it equally symmetric? Another way of phrasing it is this: what is missing?