2.15 Antiderivatives and Initial Value Problems

Summary

The concept of an antiderivative of a function f is introduced as a function F such that F' = f. Thus, finding an antiderivative can be thought of as solving the differential equation F' = f for F. Differential equations are at the heart of modelling motion in dynamic systems. They provide the language in which we can describe the state of a physical system. For example an object in motion may be described in terms of its position, velocity, and acceleration. An equation relating these properties is thus an equation involving a function and its first and second derivatives. Also introduced in this section are Initial-value problems where additional conditions are present that allow a particular solution of a differential equation to be picked out from the general solution.

By the end of your studying, you should know:

• The definition of an antiderivative.
• Any two antiderivatives of a function on an interval differ by a constant.
• The definition of an indefinite integral.
• How to find indefinite integrals.
• How to solve introductory first-order and second-order differential equations.
• How to recognize an Initial Value Problem (IVP).
• How to solve an Initial Value Problem (IVP) of first or second order.

On-screen applet instructions: Use the slider to display solutions of the differential equation.

Examples

Graph sin2(x), then use the graph to sketch an antiderivative of sin2(x).

Let g'''(x) = sin(x), with the conditions g(0) = 0, g'(0) = 1, g''(0) = –1. Find g(x).

Find a solution to the 4th-order differential equation

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Exercises

See Exercises for 2.15 Antiderivatives and Initial Value Problems (PDF).

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel