Prove that the given functions are linearly independent or dependent on the indicated interval.?

Question

f(x)=x^2, g(x)=(x^3)(e^x),
h(x) = (x^2)(e^x) ; the real line.

differential equations problem.
Please show how you got it. Thank you.

Answer 1

To show this you need to compute the Wronskian
of the 3 functions. For the theory you need to see
http://en.wikipedia.org/wiki/Wronskian
If the Wronskian is not identically 0 the functions
are linearly independent.
Please check my work, but I found that the
Wronskian of the 3 functions is the determinant
x² x³e^x x²e^x
2x x³e^x + 3x²e^x x²e^x+2xe^x
2 x³e^x + 6x²e^x + 6xe^x x²e^x + 4xe^x + 2e^x
and when I worked it out I got
-x^6e^2x, which is not identically 0.
That means the 3 functions f, g and h
are linearly independent on the real line.

Answer 2

you could have said "Prove that the given functions are linearly independent or dependent on the real line", not only copy and paste from textbook.....