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

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Math 334 Lecture #8 § 2.6: Exact First Order ODE’s 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 ψ ( x, y ) = x 2 + xy 2 are ∂ψ ∂x = 2 x + y 2 and ∂ψ ∂y = 2 xy, the ODE can be written in the form ∂ψ ∂x + ∂ψ ∂y dy dx = 0 . By the Chain Rule, the ODE simplifies to d dx ψ ( x, y ) = 0 , integration of which gives ψ ( x, y ) = C as a one parameter family of (implicitly defined) solutions of the ODE. Geometrically, the level curves of ψ 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 ψ ( x, y ) such that M ( x, y ) = ∂ψ ∂x and N ( x, y ) = ∂ψ ∂y . Solutions of an exact first order ODE are given by ψ ( x, y ) = C . The presence of the arbitrary constant C means that IVP’s may be solved.
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