**PROOF OF THEOREM 1 Uniqueness**^{1}

Assuming that the problem consisting of the ODE

and the two initial conditions

has two solutions *y*_{1}(*x*) and *y*_{2}(*x*) on the interval *I* in the theorem, we show that their difference

is identically zero on *I;* then *y*_{1} ≡ *y*_{2} on *I*, which implies uniqueness.

Since (1) is homogeneous and linear, *y* is a solution of that ODE on *I*, and since *y*_{1} and *y*_{2} 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 ...

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