Larson 11 Chapter 16.pdf - n Exact First-Order Equations...

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CYAN MAGENTA YELLOW BLACK Larson Texts, Inc. • Multivariable Calculus 11e • CALC11-WFH 16.1 Exact First-Order Equations 16.2 Second-Order Homogeneous Linear Equations 16.3 Second-Order Nonhomogeneous Linear Equations 16.4 Series Solutions of Differential Equations Electrical Circuits (Exercises 29 and 30, p. 1151) Parachute Jump (Section Project, p. 1152) Motion of a Spring (Example 8, p. 1142) Cost (Exercise 45, p. 1136) Undamped or Damped Motion? (Exercise 53, p. 1144) 1129 16 9781337275378_1600 09/13/16 Final Pages Clockwise from top left, Phovoir/Shutterstock.com; Danshutter/Shutterstock.com; ICHIRO/Photodisc/Getty Images; SasinTipchai/Shutterstock.com; photo credit to come Additional Topics in Differential Equations © Cengage Learning. Not for distribution.
1130 Chapter 16 Additional Topics in Differential Equations 16.1 Exact First-Order Equations Solve an exact differential equation. Use an integrating factor to make a differential equation exact. Exact Differential Equations In Chapter 6, you studied applications of differential equations to growth and decay problems. You also learned more about the basic ideas of differential equations and studied the solution technique known as separation of variables. In this chapter, you will learn more about solving differential equations and using them in real-life applications. This section introduces you to a method for solving the first-order differential equation M ( x , y ) dx + N ( x , y ) dy = 0 for the special case in which this equation represents the exact differential of a function z = f ( x , y ) . Definition of an Exact Differential Equation The equation M ( x , y ) dx + N ( x , y ) dy = 0 is an exact differential equation when there exists a function f of two variables x and y having continuous partial derivatives such that f x ( x , y ) = M ( x , y ) and f y ( x , y ) = N ( x , y ) . The general solution of the equation is f ( x , y ) = C . From Section 13.3, you know that if f has continuous second partials, then M y = 2 f y x = 2 f x y = N x . This suggests the following test for exactness. THEOREM 16.1 Test for Exactness Let M and N have continuous partial derivatives on an open disk R . The differential equation M ( x , y ) dx + N ( x , y ) dy = 0 is exact if and only if M y = N x . Every differential equation of the form M ( x ) dx + N ( y ) dy = 0 is exact. In other words, a separable differential equation is actually a special type of an exact equation. Exactness is a fragile condition in the sense that seemingly minor alterations in an exact equation can destroy its exactness. This is demonstrated in the next example. 9781337275378_1601 09/13/16 Final Pages CYAN MAGENTA YELLOW BLACK Larson Texts, Inc. • Multivariable Calculus 11e • CALC11-WFH © Cengage Learning. Not for distribution.
16.1 Exact First-Order Equations 1131 Testing for Exactness Determine whether each differential equation is exact.

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