I will call the Wronskian matrix To start, we should have in other words the entry in the fourth row, fourth column should be not I'm guessing you actually have this and missed a minus sign in your post.
To make life simpler, using the fact that the determinant is not affected when you add a row to another will make things easier. For example, adding row 1 to row 3 makes row 3 become If you use this fact a couple of times you should be able to get a row that has 3 zeros and one nonzero entry. You then expand your determinant about this nonzero entry and go from there.
Give it another shot with this in mind. I ended up getting when all was said and done. Since the vectors are linearly independent. Let me know if you still get stuck. Good luck!