2.6 - Math 334 Lecture #8 2.6: Exact First Order ODEs The...

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Unformatted text preview: Math 334 Lecture #8 2.6: Exact First Order ODEs The first order ODE 2 x + y 2 + 2 xy dy dx = 0 is not linear, nor is it separable. Notice, however, that since the partial derivatives of ( x,y ) = x 2 + xy 2 are x = 2 x + y 2 and y = 2 xy, the ODE can be written in the form x + y dy dx = 0 . By the Chain Rule, the ODE simplifies to d dx ( x,y ) = 0 , integration of which gives ( x,y ) = C as a one parameter family of (implicitly defined) solutions of the ODE. Geometrically, the level curves of are integral curves of the ODE. [There is no guarantee this is one parameter family of solutions is a general solution.] A first order ODE of the form M ( x,y ) + N ( x,y ) dy dx = 0 or M ( x,y ) dx + N ( x,y ) dy = 0 is called exact if there is a continuously differentiable function ( x,y ) such that M ( x,y ) = x and N ( x,y ) = y ....
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2.6 - Math 334 Lecture #8 2.6: Exact First Order ODEs The...

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