Question

(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.

Answer 1

W ={(x) € Cº: F"(x)=- } (x)} (a) consider a sin x+bcos x = 0) for all x particularly take x = 0 a sin x+bcos x=0=b=0 a sin x+bcos x=0= asin x=0 for all x a=0 therefore, (sin x, cos x} is linearly independent set. (b) dim(W)= 2 S =(sin x, cosx} is linearly independent set and n(S) = dim(W)=2 S is basis of W. (C)/"(x)=-f(x), f(0) = 2, f'(0) = 4 F"(x)=-f(x)=f(x) € W S ={sin x, cos x} is basis of W. f(x) = a cos x+b sin x f(0) = 2=a=2, f(x) = 2 cos x+b sinx f'(x) = -2 sin x+bcos x f(0) = 4= f'(x) = b = 4 :: f(x) = 2 cos x+4 sin x but f(-x) = 2cos x-4 sin x + f(x) is not an even function. :: no such function exist. note f is event is odd = f'(0) = 0