Say a function u(x) satisfies the differential equation
 (1) 
on the interval [0,A] and that the coefficients b(x) and c(x) are both bounded, say b(x)≤ M and c(x)≤ M (if the
coefficients are continuous, this is always true for some M).

a)
 Define E(x) := (u′^{2} + u^{2}). Show that for some constant γ (depending on M) we have E′(x) ≤ γE(x).
[Suggestion; use the inequality 2xy ≤ x^{2} + y^{2}.]

b)
 Use Problem 3(c) above to show that E(x) ≤ e^{γx}E(0) for all x ∈ [0,A].

c)
 In particular, if u(0) = 0 and u′(0) = 0, show that E(x) = 0 and hence u(x) = 0 for all x ∈ [0,A]. In
other words, if u′′ + b(x)u′ + c(x)u = 0 on the interval [0,A] and that the functions b(x) and c(x) are both
bounded, and if u(0) = 0 and u′(0)  0, then the only possibility is that u(x) ≡ 0 for all x ≥ 0.

d)
 Use this to prove the uniqueness theorem : if v(x) and w(x) both satisfy equation (1) and have the same
initial conditions, v(0) = w(0) and v′(0) = w′(0), then v(x) ≡ w(x) in the interval [0,A].