First-order linear equations

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 • Sol.

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