Exact Differential Equations

1Introduction

This lecture explains how to reconstruct the differential form

as the total differential of a function $$F(x,y)$$F(x,y):

Separation of variables divides an equation into an $$x$$x-dependent side and a $$y$$y-dependent side. The integrating-factor method for first-order linear equations converts the left-hand side into the derivative of a product. Exact differential equations instead represent the entire left-hand side as the total differential of one function $$F(x,y)$$F(x,y).

2Standard Form

An exact differential equation is an equation

for which there exists a function $$F(x,y)$$F(x,y) satisfying

The function $$F$$F is called a potential function.

3Proposition 1: Solutions of an Exact Equation Have the Form $$F(x,y)=C$$F(x,y)=C

Suppose that $$Mdx+Ndy=dF$$M\,dx+N\,dy=dF. Then the solutions of

are represented implicitly by

3.1Proof

Since $$Mdx+Ndy=dF$$M\,dx+N\,dy=dF, the equation is $$dF=0$$dF=0. Along a differentiable curve $$y=y\left(x\right)$$y=y(x), the chain rule gives

Substitution of $$dy=y^{\prime }dx$$dy=y'\,dx into

gives

Thus $$dF=0$$dF=0 implies

Consequently, $$F$$F is constant along each solution curve, so $$F(x,y)=C$$F(x,y)=C. Conversely, differentiation of $$F(x,y)=C$$F(x,y)=C gives $$dF=0$$dF=0 and therefore recovers the original equation.

4Exactness Condition

If $$M$$M and $$N$$N are continuously differentiable and $$F_x=M$$F_x=M and $$F_y=N$$F_y=N, then the mixed partial derivatives of $$F$$F are continuous, so

Therefore,

is necessary for exactness.

If $$M$$M and $$N$$N are continuously differentiable and the domain is simply connected, then

is also sufficient for exactness. A simply connected domain is a domain in which every closed curve can be continuously contracted to a point without leaving the domain. Rectangles and disks are standard examples.

5Proposition 2: Necessity of the Exactness Condition

Suppose that $$M$$M and $$N$$N are continuously differentiable and that $$M=F_x$$M=F_x and $$N=F_y$$N=F_y. Then

5.1Proof

Since $$M=F_x$$M=F_x,

Since $$N=F_y$$N=F_y,

Continuity of the mixed partial derivatives gives $$F_{xy}=F_{yx}$$F_{xy}=F_{yx}, and hence

Thus the exactness condition follows from the interchangeability of the mixed partial derivatives of the potential function.

6Constructing the Potential Function

Once exactness has been established, reconstruct $$F$$F from $$F_x=M$$F_x=M:

1. Integrate $$F_x=M$$F_x=M with respect to $$x$$x.
2. Add an arbitrary function $$h\left(y\right)$$h(y), because integration with respect to $$x$$x leaves the $$y$$y-dependent part undetermined.
3. Compare $$F_y$$F_y with $$N$$N to determine $$h\left(y\right)$$h(y).

The term $$h\left(y\right)$$h(y) is necessary because its partial derivative with respect to $$x$$x is zero.

7Example 1: An Exact Equation

Consider

Here,

The exactness condition holds because

To reconstruct the potential, integrate

with respect to $$x$$x:

Differentiation with respect to $$y$$y gives

Comparison with $$N=x^2$$N=x^2 yields

Thus $$h$$h is constant, and the solution is

8When an Integrating Factor Is Required

Even when

is not exact, multiplication by a function $$\mu (x,y)$$\mu(x,y) that is nonzero everywhere on the domain may produce the exact equation

Such a function $$\mu$$\mu is also called an integrating factor. Because $$\mu$$\mu is nonzero everywhere, the equations before and after multiplication represent the same solution curves.

For a first-order linear equation, an integrating factor converts the left-hand side into the product derivative

For a differential form, an integrating factor converts $$Mdx+Ndy$$M\,dx+N\,dy into a total differential

Both uses transform an equation into a directly integrable differential.

9Connection with First-Order Linear Equations

Rewrite

as

Then

In general,

so the form is not exact. Multiplication by the linear integrating factor

gives

For the transformed form,

so it is exact. Hence the integrating factor for a first-order linear equation is also an exactness-producing factor in the framework of differential forms.

10Scope

The sufficiency of $$M_y=N_x$$M_y=N_x depends on the domain. On a domain that is not simply connected, this equality may hold even though no global potential function exists.

This topological qualification reappears in the study of line integrals and conservative vector fields.

The next lecture treats equations whose right-hand side depends only on the ratio $$y/x$$y/x and explains how the substitution $$v=y/x$$v=y/x reduces them to separable equations.

11Exercises