# (5) Let W denote the set of smooth functions f(2) in CⓇ such that f'(x) =...

(5) Let W denote the set of smooth functions f(2) in CⓇ such that f'(x) = -f(L). That is, W= {f() in C | F"(x) = -f(x)} In the previous worksheet, we showed that: • W is a subspace of Cº. . For all a and b, a sin(2) + b cos(x) is in W. (a) Show that (sin(x), cos(x)} are linearly independent. Hint: Set an arbitrary linear combination equal to 0, and show the coefficients must be 0. (b) Let's say we knew that dim(W) = 2. Show that (sin(x), cos(2)} is a basis for W. (c) Find every function f(x) such that f'(2) = -f(2), f(0) = 2, and f'(0) = 4, or show that no such function exists.

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