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2.2 - Math 334 Lecture#4 2.2 Separable First Order ODEs The...

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Math 334 Lecture #4 § 2.2: Separable First Order ODE’s The ODE’s 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. [It may not be a general solution, i.e., it may not contain all possible solutions of the ODE, but it can be useful for solving IVP’s.] [We will consider in a moment an example where the one parameter family of solutions does not contain all possible solutions!]
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