In this exercise we will show that the functions cos(*) and sin(jc) span the solution

Chapter 4, Problem 58

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In this exercise we will show that the functions cos(*) and sin(jc) span the solution space V of the differential equation f"(x) = f(x). (See Example 1 of this section.) a. Show that if #(jr) is in V, then the function (g{x)) +(s'C*))2 is constant. Hint: Consider the derivative. b. Show that if g(;t) is in V, with #(0) = #'(0) = 0, then g(jt) = 0 for all x. c. If / (jc) is in V, then g(jc) = f(x) f (0) cos(x) / '( O)sin(jc) is in V as well (why?). Verify that (0) = 0 and #'(0) = 0. We can conclude that g(x) = 0 for all jc, so that /(jc) = / ( O)cos(x) + /'(0) sin(jc). It follows that the functions cos(x) and sin(jc) span V, as claimed.