(a) Show that the functions f1(x) = er1x , f2(x) = er2x , f3(x) = er3x have Wronskian W[

Chapter 4, Problem 44

(choose chapter or problem)

(a) Show that the functions f1(x) = er1x , f2(x) = er2x , f3(x) = er3x have Wronskian W[ f1, f2, f3](x) = e(r1+r2+r3)x 111 r1 r2 r3 r 2 1 r 2 2 r 2 3 = e(r1+r2+r3)x (r3 r1)(r3 r2)(r2 r1), and hence determine the conditions on r1,r2,r3 such that { f1, f2, f3} is linearly independent on every interval. (b) More generally, show that the set of functions {er1x , er2 x ,..., ern x } is linearly independent on every interval if and only if all of the ri are distinct. [Hint: Show that the Wronskian of the given functions is a multiple of the n n Vandermonde determinant, and then use in Section 3.3.]