1. Math 3120 Differential Equations with Boundary Value Problems
Chapter 2: First-Order Differential Equations
Section 2-4: Exact Equations

2. Exact Equations Consider a first order ODE of the form
Suppose there is a function  such that
and such that (x,y) = c defines y = (x) implicitly. Then
and hence the original ODE becomes
Thus (x,y) = c defines a solution implicitly.
In this case, the ODE is said to be exact.

3. Theorem 2.4.1 Suppose an ODE can be written in the form
where the functions M, N, My and Nx are all continuous in the rectangular region R: (x, y)  (,  ) x (,  ). Then Eq. (1) is an exact differential equation iff
That is, there exists a function  satisfying the conditions
iff M and N satisfy Equation (2).

4. Method to solve Exact Equations
Write the DE in the form
Check if the eq. (1) is an exact differential equation.
Write down the system of eq.
Integrate either the first equation with respect of the variable x or the second with respect of the variable y. The choice of the equation to be integrated will depend on how easy the calculations are. Let us assume that the first equation was chosen, then we get
Use the second equation of the system to find the derivative of g(y).
Integrate to find
Write down the function
All the solutions are given by the implicit equation
If you are given an IVP, plug in the initial condition to find the constant C.

5. Example 1: Exact Equation (1 of 4)
Consider the following differential equation.
Then
and hence
From Theorem 2.4.1,
Thus

6. Example 1: Solution (2 of 4)
We have
and
It follows that
Thus
By Theorem 2.6.1, the solution is given implicitly by

7. Example 1: Direction Field and Solution Curves (3 of 4)
Our differential equation and solutions are given by
A graph of the direction field for this differential equation,
along with several solution curves, is given below.

8. Example 1: Explicit Solution and Graphs (4 of 4)
Our solution is defined implicitly by the equation below.
In this case, we can solve the equation explicitly for y:
Solution curves for several values of c are given below.

9. Example 2: Exact Equation (1 of 3)
Consider the following differential equation.
Then
and hence
From Theorem 2.4.1,
Thus

10. Example 2: Solution (2 of 3)
We have
and
It follows that
Thus
By Theorem 2.6.1, the solution is given implicitly by

11. Example 2: Direction Field and Solution Curves (3 of 3)
Our differential equation and solutions are given by
A graph of the direction field for this differential equation,
along with several solution curves, is given below.

12. Example 3: Non-Exact Equation
Consider the following differential equation.
Then
and hence

13. Integrating Factors
It is sometimes possible to convert a differential equation that is not exact into an exact equation by multiplying the equation by a suitable integrating factor (x, y):
For this equation to be exact, we need
This partial differential equation may be difficult to solve. If  is a function of x alone, then y = 0 and hence we solve
provided right side is a function of x only. Similarly if  is a function of y alone.

14. Summarize Given a first ODE by
If is a function of x alone, then an integrating factor for (1) is
If is a function of y alone, then an integrating factor for (1) is

15. Example 4: Non-Exact Equation
Consider the following non-exact differential equation.
Seeking an integrating factor, we solve the linear equation
Multiplying our differential equation by , we obtain the exact equation
which has its solutions given implicitly by