PROOF OF THEOREM 1 Uniqueness1
Assuming that the problem consisting of the ODE
and the two initial conditions
has two solutions y1(x) and y2(x) on the interval I in the theorem, we show that their difference
is identically zero on I; then y1 ≡ y2 on I, which implies uniqueness.
Since (1) is homogeneous and linear, y is a solution of that ODE on I, and since y1 and y2 satisfy the same initial conditions, y satisfies the conditions
We consider the function
and its derivative
From the ODE we have
By substituting this in the expression for z′ we obtain
Now, since y and y′ are real,
From this and the definition of z we obtain the two inequalities ...