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Advanced Engineering Mathematics, 10th Edition by Edward J. Norminton, Herbert Kreyszig, Erwin Kreyszig

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APPENDIX 4

Additional Proofs

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Section 2.6, page 74

PROOF OF THEOREM 1 Uniqueness1

Assuming that the problem consisting of the ODE

image

and the two initial conditions

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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

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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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