Copyright © Cengage Learning. All rights reserved. 7 Further Integration Techniques and Applications of the Integral

Copyright © Cengage Learning. All rights reserved. 7.6 Differential Equations and Applications

3 A differential equation is an equation that involves a derivative of an unknown function. A first-order differential equation involves only the first derivative of the unknown function. A second-order differential equation involves the second derivative of the unknown function (and possibly the first derivative). Higher order differential equations are defined similarly.

4 Differential Equations and Applications To solve a differential equation means to find the unknown function. Many of the laws of science and other fields describe how things change. When expressed mathematically, these laws take the form of equations involving derivatives—that is, differential equations.

5 Example 1 – Motion A dragster accelerates from a stop so that its speed t seconds after starting is 40t ft/s. How far will the car go in 8 seconds? Solution: We wish to find the car’s position function s(t). We are told about its speed, which is ds/dt. Precisely, we are told that This is the differential equation we have to solve to find s(t). But we already know how to solve this kind of differential equation; we integrate

6 Example 1 – Solution We now have the general solution to the differential equation. By letting C take on different values, we get all the possible solutions. We can specify the one particular solution that gives the answer to our problem by imposing the initial condition that s(0) = 0. Substituting into s(t) = 20t 2 + C, we get 0 = s(0) so C = 0 and s(t) = 20t 2. To answer the question, the car travels 20(8) 2 = 1,280 feet in 8 seconds. cont’d = 20(0) 2 + C = C

7 Differential Equations and Applications Simple Differential Equations A simple differential equation has the form Its general solution is

8 Differential Equations and Applications Quick Example The differential equation is simple and has general solution

9 Differential Equations and Applications Separable Differential Equation A separable differential equation has the form We solve a separable differential equation by separating the xs and the ys algebraically, writing and then integrating: