Milestones_4_Math_33B_F08

Milestones_4_Math_33B_F08 - Mathematics 33B: Milestones for...

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Mathematics 33B: Milestones for Week 4 4.1 Existence and uniqueness, linear dependence, the Wronskian, 4.3 sec- ond order constant coefficient equations. This week is not on this Midterm. Lecture 11: Second-Order Equations. A. Basic Definition: A second order differential equation has the form y 00 = f ( t, y, y 0 ). 1. A solution to such an equation is a twice continuously differentiable function y = y ( t ) with y 00 ( t ) = f ( t, y ( t ) , y 0 ( t )). 2. Linear Equations. They have the form y 00 + p ( t ) y 0 + q ( t ) y = g ( t ). Example: F = ma is linear - Newton’s Force law. Example: θ 00 = k sin θ is not - pendulum. Example: Vibrating Spring example. We postpone most of the discussion. B. Existence and uniqueness: if p, q, g are continuous on ( α, β ) then there is one and only one y ( t ) defined on ( α, β ) with y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) and such that y ( t 0 ) = y 0 and y 0 ( t 0 ) = y 1 . C. A little linear algebra review. What is a vector space? What is a
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Milestones_4_Math_33B_F08 - Mathematics 33B: Milestones for...

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