1. 6.1: DIFFERENTIAL EQUATIONS AND SLOPE FIELDS

2. DEFINITION: DIFFERENTIAL EQUATION An equation involving a derivative is called a differential equation.

3. SOLVING A DIFFERENTIAL EQUATION

4. SOLVING AN INITIAL VALUE PROBLEM

5. EXAMPLE 3:

6. EXAMPLE 4:

7. SLOPE FIELDS: A graphical representation of the solutions of a differential equation. It is useful because it can be created without solving the differential equation analytically. Slope fields provide an excellent way to visualize a family of solutions of differential equations. The slope field provides a way to solve the equation graphically. Slope fields also give us a great way to visualize a family of antiderivatives.

8. SLOPE FIELDS x0π/2π-π/2-π-π3π/2-3π/2 dy/dx

9. x000111 y 01 01 dy/dx01

10. WHAT TO LOOK FOR WHEN DEALING WITH SLOPE FIELDS: 1.Look for places where the slope is 0. 2.Look at the slope along the x-axis. 3.Look at the slope along the y-axis. 4.Look to see if the slopes only depend on x. 5.Look to see if the slopes only depend on y. 6.Look to see where the slopes are positive and negative.

11. Do example 8 on page 325 (without looking at solution).