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exact differential equation

type of differential equation that can be solved directly without the use of any of the special techniques in the subject. A first-order differential equation (of one variable) is called exact, or an exact differential, if it is the result of a simple differentiation. The equation P(x,y)y' + Q(x,y)=0, or in the equivalent alternate notation P(x,y)dy+Q(x,y)dx=0, is exact if Px(x,y)=Qy(x,y). (The subscripts in this equation indicate which variable the partial derivative is taken with respect to.) In this case, there will be a function R(x,y), the partial x-derivative of which is Q and the partial y-derivative of which is P, such that the equation R(x,y)=c (where c is constant) will implicitly define a function y that will satisfy the original differential equation.

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Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.
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