Lecture 5 - Exact Equations Nonlinear Non-separable First...

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Exact Equations Nonlinear Non-separable First Order Equations
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Techniques We Know m ( y ) y 0 = n ( x ) y 0 + p ( t ) y = q ( t ) Nonlinear Separable Separation of Variables Linear Integrating Factors M ( x, y ) + N ( x, y ) y 0 = 0 Nonlinear Non-separable
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Implicit Differentiation 5 x 2 y 3 + 7 x 3 ( ) 0 = 10 xy 3 21 x 2 15 x 2 y 2 + + y 0 @ @ x ( 5 x 2 y 3 + 7 x 3 ) = 10 xy 3 + 21 x 2 @ @ y ( 5 x 2 y 3 + 7 x 3 ) = 15 x 2 y 2 Note:
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Implicit Differentiation 5 x 2 y 3 + 7 x 3 ( ) 0 = 10 xy 3 21 x 2 15 x 2 y 2 + + y 0 @ @ x ( 5 x 2 y 3 + 7 x 3 ) = 10 xy 3 + 21 x 2 @ @ y ( 5 x 2 y 3 + 7 x 3 ) = 15 x 2 y 2 f ( x, y ) 0 = f x ( x, y ) + f y ( x, y ) y 0
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Implicit Differentiation f ( x, y ) 0 = f x ( x, y ) + f y ( x, y ) y 0 This is the general rule: This comes from the terms with x This comes from the terms with y
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Implicit Differentiation This is the general rule: So to solve problems of the form: M ( x, y ) + N ( x, y ) y 0 = 0 f ( x, y ) 0 = f x ( x, y ) + f y ( x, y ) y 0 What do we do? Find: f ( x, y ) Where: f x ( x, y ) = M ( x, y ) f y ( x, y ) = N ( x, y )
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Solving Nonlinear Non-Separable Equations To solve problems of the form: M ( x, y ) + N ( x, y ) y 0 = 0 Find: f ( x, y ) Where: f x ( x, y ) = M ( x, y ) f y ( x, y ) = N ( x, y ) We’ll talk about how to do this in a minute
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