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 (analysis). 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 P_x(x,y)=Q_y(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.

For example, in the equation (x²+2y)y′+2xy+1=0, the x-derivative of x²+2y is 2x and the 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 (logarithm) (ln) appears: 3lnx+2lny=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)y″+q(x)y′+r(x)y=0 is exact if there is a first-order expression p(x)y′+s(x)y such that its derivative is the given equation. The given equation will be exact if, and only if, p″−q′+r=0, in which case s in the reduced equation will equal q−p′. 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.