Exact Equation

exact 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)^dy/_dx + Q(x, y) = 0,or in the equivalent alternate notation P(x, y)dy + Q(x, y)dx = 0,is exact if ^∂P(x, y)/_∂x = ^∂Q(x, y)/_∂y.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.

For example, in the equation (x² + 2y)^dy/_dx + 2xy + 1 = 0,the partial x-derivative of x² + 2y is 2x and the partial y-derivative of 2xy + 1 is also 2x, and the function R = x²y + x + y² satisfies the conditions R_x = Q and R_y = P. The function defined implicitly by x²y + x + y² = c will solve the original equation. Sometimes if an equation is not exact, it can be made exact by multiplying each term by a suitable function called an integrating factor. For example, if the equation 3y + 2xy′ = 0 is multiplied by 1/xy, it becomes 3/x + 2y′/y = 0, which is the direct result of differentiating the equation in which the natural logarithmic function (ln) appears: 3 ln x + 2 ln y = c, or equivalently x³y² = c, which implicitly defines a function that will satisfy the original equation.

Higher-order equations are also called exact if they are the result of differentiating a lower-order equation. For example, the second-order equation p(x)^d2y/_dx² + q(x)^dy/_dx + r(x)y = 0is exact if there is a first-order expressionp(x)^dy/_dx + s(x)ysuch that its derivative is the given equation. The given equation will be exact if, and only if,p^d2y/_dx² + q^dy/_dx + r = 0,in which case s in the reduced equation will equal q − p(^dy/_dx). If the equation is not exact, there may be a function z(x), also called an integrating factor, such that when the equation is multiplied by the function z it becomes exact.