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To test this possibility, we first write the equation in the operator form (1) Now (D2 + a1D + a2)erx = D2(erx) + a1D(erx) + a2 erx = r2erx + a1rerx + a2erx = erx (r 2 + a1r + a2) The latter expression is zero, provided (2) r2 + a1r + a2 = 0 Equation (2) is called the auxiliary equation for (1) (note the similarity in form). be a solution to our differential equation for an appropriate choice of r. Moreover, it can be shown that every solution has this form. By the linearity of the operator D2 + a1 D + a2, C1u1(x) + C2 u2(x) is also a solution. In fact, a second-order homogeneous linear differential equation always has two fundamental solutions u1(x) and u2(x), which are independent of each other (i.e., neither function is a constant multiple of the other). By analogy, we might expect that solving a second-order equation to involve two integrations and thus that the general solution would have two arbitrary constants. To solve a first-order equation required one integration and led to a general solution with one arbitrary constant. Thus, our initial task is to solve y + a1 y + a2 y = 0 A differential equation for which k(x) = 0 is said to be homogeneous. In this section, we make two simplifying assumptions: (1) a1(x) and a2(x) are constants, and (2) k(x) is identically zero. Second-Order Linear Equations A second-order linear differential equay + a1(x)y + a2(x)y = k(x) The theory of nonlinear differential equations is both complicated and fascinating, but best left for more advanced courses.Д§76 Chapter 15 Differential Equationstion has the form The presence of the exponent 2 on y is enough to spoil the linearity, as you may check. Many important differential equations, such as dy + y2 = 0 dx are nonlinear. Of course, not all differential equations are linear. Thus, if L denotes this operator and if f and g are functions and c is constant, then L(f + g) = L(f) + L(g) L(cf) = cL(f) That L has these properties follows readily from the corresponding properties for the derivative operators D, D2,, Dn. Then the operator in brackets is a linear operator.

(Note that y and all its derivatives occur to the first power.) This is called a linear equation because, if it is written in operator notation,n C Dn + a1(x)Dx - 1 + + an - 1(x)Dx + an(x) D y = k(x) x In this chapter, we consider only nth-order linear differential equations, that is, equations of the form y (n) + a1(x)y(n - 1) + + an - 1(x)y + an(x)y = k(x) where n 2. In Section 6.5, we solved the differential equation y = ky of exponential growth and decay, and in Section 6.6 we studied first-order linear differential equations and some applications. In Section 3.9, we introduced the technique called separation of variables and used it to solve a wide variety of first-order equations. Differential equations appeared earlier in this book, principally in three sections. In contrast, 2 cos x + 10 is called a particular solution of the equation. We call 2 cos x + C the general solution of the given equation, since it can be shown that every solution can be written in this form. Thus, f(x) = 2 cos x + 10 is a solution to y + 2 sin x = 0 since f(x) + 2 sin x = -2 sin x + 2 sin x = 0 for all x.

If, when f(x) is substituted for y in the differential equation, the resulting equation is an identity for all x in some interval, then f(x) is called a solution of the differential equation. Examples of differential equations of orders 1, 2, and 3 are y + 2 sin x = 0 dy dx dx2 2ДЎ5.1 Linear Homogeneous Equations 15.2 Nonhomogeneous Equations 15.3 Applications of Second-Order Equations In particular, an equation of the form F (x, y, y(1), y(2),, y(n)) = 0 in which y(k) denotes the kth derivative of y with respect to x, is called an ordinary differential equation of order n. Differential EquationsLinear Homogeneous EquationsWe call an equation involving one or more derivatives of an unknown function a differential equation.

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