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Unformatted text preview: Math 334 Lecture #4 2.2: Separable First Order ODEs The ODEs dy dx =- 1 5 y + 9 . 8 and y = xy 2 1 + x 2 can be rewritten respectively as- 1 + 1- y/ 5 + 9 . 8 dy dx = 0 and- x 1 + x 2 + 1 y 2 y = 0 . [The first ODE is linear; the second ODE is nonlinear.] Each of these is an example of a separable first order ODE, the general form of which is M ( x ) + N ( y ) dy dx = 0 for functions M ( x ) and N ( y ). Method of Solution. Let H ( x ) and G ( y ) be antiderivatives of M ( x ) and N ( y ): d dx H ( x ) = M ( x ) and d dy G ( y ) = N ( y ) . The general form of the separable first order ODE becomes d dx H ( x ) + d dy G ( y ) dy dx = 0 . By the Chain Rule, the second term is d dx G ( y ) = d dy G ( y ) dy dx , so that the ODE becomes d dx H ( x ) + d dx G ( y ) = 0 d dx H ( x ) + G ( y ) = 0 . Integration gives H ( x ) + G ( y ) = C as a one parameter family of solutions of the separable first order ODE....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11