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By Peter V O'Neil

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Y Computing ∂ϕ/∂ x is less straightforward, since x occurs in both integrals defining ϕ(x, y). For ∂ϕ/∂ x, use the condition that ∂ M/∂ y = ∂ N /∂ x to write ∂ ∂ϕ = ∂x ∂x x M(ξ, y0 ) dξ + x0 y = M(x, y0 ) + y0 y = M(x, y0 ) + y0 ∂ ∂x y N (x, η) dη y0 ∂N (x, η) dη ∂x ∂M (x, η) dη ∂y = M(x, y0 ) + M(x, y) − M(x, y0 ) = M(x, y). This completes the proof. In the case of y d x + dy = 0, M(x, y) = y and N (x, y) = 1, so ∂N ∂M = 1 and = 0. 1 tells us that this differential equation is not exact on any rectangle in the plane.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 9 Two families of orthogonal trajectories.

Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4 Homogeneous, Bernoulli, and Riccati Equations 27 or x du = f (u) − u. dx The variables u and x separate as 1 1 du = d x. f (u) − u x We attempt to solve this separable equation and then substitute u = y/x to obtain the solution of the original homogeneous equation.

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