1. Differential Equations
Separation of variables
Exact factors of Differential Equations
Integrating factors
Before you begin you should be confident with:
Integration by parts, substitution and trigonometric functions
Differential Equations by separation of variables
Implicit differentiation

2. Reminder Solve the following differential equations x=1, y=0 x=0, y=0
Integrate by parts

3. Try solving this differential equation
When x=0 and y=1
We cannot separate the variables in the above, however, you may have noticed that the LHS is the exact derivative of x2y..
We can therefore write the equation as..
Substituting x=0, y=1

4. Solve the following differential equation…
You may notice that…
When x=1 and y=1

5. Now solve this differential equation
Exact derivatives
We were able to solve these two differential equations as the LHS were exact derivatives.
Unfortunately this isn’t always the case…
This is not an exact derivative of anything.
What could I multiply it by to make it an exact derivative??
We call “x” the integrating factor
Now solve this differential equation

7. Integrating Factors
It is not always obvious what the integrating factor is but fortunately, we can work it out.
Consider the differential equation..
Notice the coefficient of dy/dx is 1
We multiply by an integrating factor, f(x)

8. Notice the coefficient of dy/dx is 1
We multiply by an integrating factor, f(x)

9. Notice the coefficient of dy/dx is 1
We multiply by an integrating factor, f(x)
The first term is one of the terms we get when we differentiate yf(x).
We want the second term to be the other.

10. The Shizzle
This is what we use to find the integrating factor

11. Example 1 Note: The coefficient of dy/dx must be 1
P(x) is the coefficient of y
This is our integrating factor

12. Example 2
This is our integrating factor

13. Exercise 1. Find the general solution to the differential equation
2. Given that when x = 0, y = 1, solve the differential equation

14. Answers 1. Find the general solution to the differential equation
2. Given that when x = 0, y = 1, solve the differential equation

15. Links to other resources
Mathsnet Q1
Mathsnet Q2
Mathsnet Q3
Mathsnet Q4
Mathsnet Q5