Two functions u(x) and v(x) are said to be linearly dependent on an interval I if...

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Two functions u(x) and v(x) are said to be linearly dependent on an interval I if there is a constant c such that u(x) = cv(x) or v(x) = cu(x). (a) Show that if u and v are linearly dependent, then the Wronskian is W (u, v) = 0. (b) Suppose that v(x) #0 on I. Show that u d () = _W ( dx W(u, v) 12 (c) Show that if v(x) 0 and W (u, v) = 0 on I, then u and v are linearly dependent. (d) Show that W (x, |x|x) = 0 for x = (-,). Does this mean that x and xx are linearly dependent on (-,)? Sketch the functions to bolster your argument. Two functions u(x) and v(x) are said to be linearly dependent on an interval I if there is a constant c such that u(x) = cv(x) or v(x) = cu(x). (a) Show that if u and v are linearly dependent, then the Wronskian is W (u, v) = 0. (b) Suppose that v(x) #0 on I. Show that u d () = _W ( dx W(u, v) 12 (c) Show that if v(x) 0 and W (u, v) = 0 on I, then u and v are linearly dependent. (d) Show that W (x, |x|x) = 0 for x = (-,). Does this mean that x and xx are linearly dependent on (-,)? Sketch the functions to bolster your argument.

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