Exact First-order Equations 1093 Exact Differential Equations ; Integrating Factors Exact Differential Equations In Section 5. 6, you studied applications of differential equations to growth and decay problems. In Section 5. 7, you 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 offertories differential equation Mix, DXL 0 Nix, day 0 0 or 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, DXL 0 Nix, day 0 0 is an exact differential equation if there exists a function f of two variables x and y having continuous partial derivatives such that fix and x, y 0 Nix, y. The general solution of the equation is f x, y 0 C. From Section 12. , you know that if f has continuous second partials, then if HEX This suggests the following test for exactness. THEOREM 15. 1 Test for Exactness Let M and N have continuous partial derivatives on an open disc R. The differential equation M x, DXL 0 Nix, day 0 0 is exact if and only if 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 following example. 1094 CHAPTER 15 Differential Equations EXAMPLE 1 Testing for Exactness a.
The differential equation’ xx 2 0 DXL 0 yes 2 day 0 0 is exact because NOTE Every differential equation of the form x-YOU]xx-y is exact. In other words, a separable variables equation is actually a special type of an exact equation. EX.. But the equation’ y 2 0 DXL 0 xx day 0 0 is not exact, even though it is obtained by viding both sides of the first equation by x. B. The differential equation coos y DXL CT y 2 0 x sin day 0 0 is exact because coos Sin y hissings Sin y. But the equation coos y DXL CT y 2 0 x sin day 0 0 is not exact, even though it differs from the first equation only by a single sign.
Note that the test for exactness of Mix, DXL 0 Nix, day 0 0 is the same as the test for determining whether Fix,y 0 M x, y I 0 N’ x, y J is the gradient of a potential function (Theorem 14. 1). This means that a general solution f x, y 0 C to an exact differential equation can be found by the method used to find a potential function for a noncreative vector field. EXAMPLE 2 Solving an Exact Differential Equation Solve the differential equation’ ex. 0 xx DXL CT x 2 0 29 day 0 0. Solution The given differential equation is exact because EX.!.
The general solution, f x, y C, is given by f x, 90 c = 1000 M’ x, DXL NYC]3x2dxox2yox30gy. In Section 14. 1, you determined g 9 by integrating Nix, with respect to y and reconciling the two expressions for f x, y. An alternative method is to partially differentiate this version off x, with respect to y and compare the result with Nix, y. In other words, 24 Nix, foxy,y xyox30gyox20gryox202Y. Y 20 c=100 16 12 c=10 -12 Figure 15. 1 -4 Thus, g] 0 ay, and it follows that g y 0 O 2 0 CLC . Therefore, x 4 8 f x, 90 x 2yox30Y20C1 and the general solution is x 2 y x 3 y 2 0 C.
Figure 15. 1 shows the solution curves that correspond to C 1, 10, 100, and 1000. EXAMPLE 3 TECHNOLOGY you can use a graphing utility to graph a particular solution that satisfies the initial condition of a differential equation. In Example 3, the differential equation and initial conditions are satisfied when xx 2 x coos x 0, which implies that the particular solution can be written as x 0 or you В± ‘Socio x . On a graphing calculator screen, the solution would be represented by Figure 15. 2 together with the y-axis. 095 coos x 0 x sin x ay DXL 0 ex. day 00 that satisfies the initial condition y 1 when x 0. Solution The differential equation is exact because coos x Ox sin x0yZ02y42xx. Because N x, is simpler than Mix, y, it is better to begin by integrating Nix, y. -12. 57 Nix, day 0 ex. day 0 xx 2 0 Mix, 12. 57 fix, y sexy”gyring coos x Ox sin coccyx coxswain Thus, coos x Ox sin x and coos x Ox sin I DXL which implies that f x,} Oozy x coos x CLC General solution , and the general solution is xx 2 0 x coos x Applying the given initial condition produces 2 which implies that C 0 0.