1. First-order linear equations
A first-order linear equation has the general form
If the equation is called homogeneous; otherwise
it is called inhomogeneous.
For example, is a linear equation, and an
inhomogeneous one, since it can be written as

2. Integrating factor method
To solve the first-order linear equation
we multiply the equation by a suitable function I(x):
If the factor I(x) is chosen such that
then equation (2) becomes
which can be solved by

3. Integrating factor method
Thus the key point to solve equation (1) is to find I(x) such
that equation (3) holds true:
This is equivalent to
which is a separable equation for I(x). Its solution is
Simply taking C=1, we call an integrating
factor of equation (1).

4. Example Ex. Solve the equation Sol. An integrating factor is
Multiplying I(x) to the equation, we get
Ex. Solve
Sol.

5. Example Ex. Solve the equation
Sol. Not a linear equation? What if we treat x as dependent
variable and y as independent variable:

6. Example Ex. Solve the equation Sol.
Ex. Solve the initial value problem

7. Example
Ex. Solve the initial value problem
Sol.

8. Homework 22
Section 9.3: 7, 10, 15
Section 9.6: 12, 14, 19
Page 648: 1