5.8 Show that if x, y E D and they satisfy the boundary conditions az (a)- Bz' (a) = 0, yz(b) + dz' (b) = 0, where a² + 32 > 0, y² +62 > 0, then <Lx, y>=<x, Ly > 5.9 Define the functions ch and sh to be the solutions of the differential equation r"-x=0 satisfying the initial conditions ch(0) = 1, ch'(0) = 0 and sh(0) = 0, sh'(0) = 1, respectively. State and prove a theorem like Theorem 5.13 giving properties of ch and sh. Theorem 5.13 (Properties of s and c) (i) s'(t) = c(t), c'(t) = -s(t) 5.2. AN INTERESTING EXAMPLE (ii) s(t)+c2(t) = 1 s(t)c(a)+s(a)c(t), (iii) s(t)=s(t), c(-t) = c(t) (iv) s(t+a) (v) s(t-a) = s(t)c(a) - s(a)c(t), for t, a E R. 19 c(t+a) = c(t)c(a) - s(t)s(a) c(ta) = c(t)c(a) + s(t)s(a)