notes 2-4

# notes 2-4 - problem is certain to exist t-1(9-t 2 y 4 ty =...

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SECTION 2.4 A LITTLE BIT OF THEORY You’ve created a model of a physical system that involves a diﬀerential equation for an unknown function y . This y is unknown in that we don’t have a formula for it, but it already has physical signiﬁcance. Can you say anything about whether a solution for y actually exists mathematically, without having to solve the diﬀerential equation? And if a solution does exist, how many solutions are there, mathematically? Answers to such questions can tell us something about our model! EXISTENCE AND UNIQUENESS THEOREM, LINEAR CASE Statement on page 68, Theorem 2.4.1. EXAMPLE. Determine an interval in which the solution of the following initial value

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Unformatted text preview: problem is certain to exist. ( t-1)(9-t 2 ) y + 4 ty = cos(7 t ) , y (0) = 83 Same diﬀerential equation, but with initial condition y (2) = 8. EXISTENCE & UNIQUENESS THEOREM, GENERAL FIRST ORDER CASE Statement on page 70, Theorem 2.4.2. EXAMPLE. State where in the ty-plane the hypotheses of the General First Order Exis-tence and Uniqueness Theorem are satisﬁed for the diﬀerential equation y = ln | ty | 1-t 2-y 2 EXAMPLE. Verify that both y 1 ( t ) = 1-t and y 2 ( t ) =-t 2 / 4 are solutions of the initial value problem y =-t + ( t 2 + 4 y ) 1 / 2 2 , y (2) =-1 . Discuss! HOMEWORK: SECTION 2.4...
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notes 2-4 - problem is certain to exist t-1(9-t 2 y 4 ty =...

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