L02 - Review Differential equations, Classification, Com-...

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Unformatted text preview: Review Differential equations, Classification, Com- plete Solution (Reduction to Integration) of FOLODE. Example: The FOLODE is y + 3 t · y = t n . Review Differential equations, Classification, Com- plete Solution (Reduction to Integration) of FOLODE. Example: The FOLODE is y + 3 t · y = t n . We multiply by the integrating factor e R 3 /t dt = e 3 ln t = t 3 and get t 3 ( y + 3 t · y ) | {z } = ( t 3 · y ) = t n +3 , whence t 3 · y ( t ) = t n +4 n + 4 + C and y ( t ) = t n +1 n + 4 + C t 3 . We’ll consider general FOODEs y ( t ) = f ( t,y ( t ) ) , y ( t ) = dy ( t ) dt , for a while. The function f of two vari- ables t,y is called the coupling coefficient , vis. dy ( t ) = f ( t,y ( t ) ) · dt 2 Direction Fields The slope is f ( t, y ) ( t, y ) t y A Direction Field 3 ( t, y ) t y The blue curves y = y ( t ) satisfiy y prime ( t ) = f ( t, y ( t )) The slope is f ( t, y ). at every point ( t, y ) of the t, y –plane. A Direction Field plus two solutions 4 Existence and Uniqueness Theorem: Suppose the function f ( t,y ) and its partial derivatives f t def = ∂f/∂t...
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This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas.

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L02 - Review Differential equations, Classification, Com-...

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