notes 3-2

# notes 3-2 - SECTION 3.2 A BIT OF THEORY FOR SECOND ORDER...

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SECTION 3.2 A BIT OF THEORY FOR SECOND ORDER LINEAR ODE’S To answer the questions at the end of the last section we need a bit of theory, and we may as well do this for the general second order homogeneous linear ODE y 00 + p ( t ) y 0 + q ( t ) y = 0 . Notice that the coefficient of y 00 is 1. We abbreviate the left-hand-side as L [ y ], and then we call L a differential operator. If we let D stand for “take the derivative of” and then D 2 stand for “take the second derivative of,” and so on, we can even write L = D 2 + p ( t ) D + q ( t ), though we must use this notation carefully because it mixes up algebraic multiplication and application of the operator D . EXISTENCE AND UNIQUENESS FACT. This is carefully stated on page 144. The heart of the matter is that we need continuity in an interval and initial values for both y and y 0 at the same t 0 in the interval. Then the solution of the initial value problem exists and is unique and is defined throughout the interval on which we have continuity. We do not need

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