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ExactDifferentialEqu

ExactDifferentialEqu - Exact Differential Equations...

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Exact Differential Equations Definition : Let F(x,y) be a function of two real variables such that F has continuous first partial derivatives in a domain D in the xy-plane. The total differential of F, denoted by dF, is given by the expression: dF(x,y) = ! F ! x (x,y) dx + ! F ! y (x,y) dy, for all (x,y) " D Example : Given F(x,y) = x 3 sin(y) + y 2 x, then its partial derivatives are: ! F ! x = 3x 2 sin(y) and ! F ! y = x 3 cos(y) + 2yx they are continuous functions in the whole xy-plane. Therefore, dF(x,y) = (3x 2 sin(y)) dx + (x 3 cos(y) + 2yx) dy Definition : The expression M(x,y) dx + N(x,y) dy is called an exact differential form or a conservative differential from in a domain D if there exists a function F(x,y), called the potential function , defined on D such that dF(x,y) = M(x,y) dx + N(x,y) dy for all (x,y) in D. Remark : If M(x,y) dx + N(x,y) dy is an exact differential form and F(x,y) is a potential function then ! F ! x (x,y) = M(x,y) and ! F ! y (x,y) = N(x,y) . Definition : If M(x,y) dx + N(x,y) dy is an exact differential form, then the equation M(x,y) dx + N(x,y) dy = 0 is called an exact differential equation . Example : The form 2xy dx + x 2 dy is an exact differential form, since it is the total differential of the function F(x,y) = x 2 y. Then the equation 2xy dx + x 2 dy = 0 is an exact differential equation.
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