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# set4 - \$ Set 4 First Order ODEs Part 3 Kyle A Gallivan...

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a39 a38 a36 a37 Set 4: First Order ODEs - Part 3 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1

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a39 a38 a36 a37 Nonlinear First Order ODEs Suppose we have a first order nonlinear ODE: dy dx = f ( x, y ) Note the change in notation and that we can always write this as: M ( x, y ) + N ( x, y ) dy dx = 0 Differential Form: M ( x, y ) dx + N ( x, y ) dy = 0 2
a39 a38 a36 a37 Nonlinear First Order ODEs We can treat x as the independent variable, i.e., y = φ ( x ) We can treat y as the independent variable, i.e., x = θ ( y ) What are the integral curves? When is an ODE defined for a choice of independent variable? Can we find y = φ ( x ) or x = θ ( y ) ? We consider a class of nonlinear first order ODEs that can be solved by direct integration. 3

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a39 a38 a36 a37 Separable Nonlinear First Order ODEs Definition 4.1. An ODE is separable if it can be written as M ( x ) + N ( y ) dy dx = 0 Note that it is still nonlinear since there is a product of dy/dx with y (assuming N ( y ) is nontrivial).
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set4 - \$ Set 4 First Order ODEs Part 3 Kyle A Gallivan...

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