1. Slide 6- 1 What you’ll learn about Differential Equations Slope Fields Euler’s Method … and why Differential equations have been a prime motivation for the study of calculus and remain so to this day.

2. First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given:findWe don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

3. If we have some more information we can find C. Given: and when, find the equation for. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

4. Slide 6- 4 Differential Equation An equation involving a derivative is called a differential equation. The order of a differential equation is the order of the highest derivative involved in the equation.

5. Slide 6- 5 Example Solving a Differential Equation

6. Slide 6- 6 First-order Differential Equation If the general solution to a first-order differential equation is continuous, the only additional information needed to find a unique solution is the value of the function at a single point, called an initial condition. A differential equation with an initial condition is called an initial-value problem. It has a unique solution, called the particular solution to the differential equation.

7. Slide 6- 7 Example Solving an Initial Value Problem

8. Slide 6- 8 Example Using the Fundamental Theorem to Solve an Initial Value Problem

9. Initial value problems and differential equations can be illustrated with a slope field. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand.

10. Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 000 010 00 00 2 3 10 2 112 204 0 -2 0-4

11. If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

12. Integrals such as are called definite integrals because we can find a definite value for the answer. The constant always cancels when finding a definite integral, so we leave it out!

13. Integrals such as are called indefinite integrals because we can not find a definite value for the answer. When finding indefinite integrals, we always include the “plus C”.