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).
Can every differential equation be written as an exact equation?
A first-order differential equation is exact if it has a conserved quantity. For example, separable equations are always exact, since by definition they are of the form: M(y)y + N(t)=0, ... so ϕ(t, y) = A(y) + B(t) is a conserved quantity.
Can a non exact differential equation be made exact?
Such a function μ is called an integrating factor of the original equation and is guaranteed to exist if the given differential equation actually has a solution. ... Integrating factors turn nonexact equations into exact ones.
