L02 - Review Dierential equations Classication Complete...

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Review Differential equations, Classification, Com- plete Solution (Reduction to Integration) of FOLODE. Example: The FOLODE is y 0 + 3 t · y = t n .
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Review Differential equations, Classification, Com- plete Solution (Reduction to Integration) of FOLODE. Example: The FOLODE is y 0 + 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 0 + 3 t · y ) | {z } = ( t 3 · y ) 0 = t n +3 ,
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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 0 ( t ) = f ( t, y ( t ) ) , y 0 ( 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
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Direction Fields The slope is f ( t, y ) ( t, y ) t y A Direction Field 3
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( 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
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Existence and Uniqueness Theorem: Suppose the function f ( t, y ) and its partial derivatives f t def = ∂f/∂t and f y def = ∂f/∂y are continuous on the Domain D of the t, y –plane.
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