Advanced Engineering Mathematics, 10th Edition

APPENDIX 4

Additional Proofs

Section 2.6, page 74

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 ...