Chapter 4.1, Problem 58E

In this exercise we will show that the functions cos(x) and sin(x) span the solution space V of the differential equation $f^{″}\left(x\right)=-f\left(x\right)$.See Example 1 of this section.
a. Show that if $g\left(x\right)$ is in V, then the function ${\left(g\left(x\right)\right)}^2+{\left(g^{\prime }\left(x\right)\right)}^2$ is constant. Hint: Consider the derivative.
b. Show that if g(x) is in V, with $g\left(0\right)=g^{\prime }\left(0\right)=0$ , then $g\left(x\right)=0$ for all x.
c. If $f\left(x\right)$ is in V, then $g\left(x\right)=f\left(x\right)-f\left(0\right)\cos \left(x\right)-f^{\prime }\left(0\right)\sin \left(x\right)$ is in V as well (why?). Verify that
$g\left(0\right)=0$ and $g^{\prime }\left(0\right)=0$.We can conclude that
$g\left(x\right)=0$ for all x, so that $f\left(x\right)=f\left(0\right)\cos x+f^{\prime }\left(0\right)\sin \left(x\right)$.It follows that the functions cos(x) and
sin(x) span V, as claimed.