APPENDIX 4

Additional Proofs

image

Section 2.6, page 74

PROOF OF THEOREM 1 Uniqueness1

Assuming that the problem consisting of the ODE

image

and the two initial conditions

image

has two solutions y1(x) and y2(x) on the interval I in the theorem, we show that their difference

image

is identically zero on I; then y1y2 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

image

We consider the function

image

and its derivative

image

From the ODE we have

image

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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