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Unformatted text preview: Chapter 8. Differential Equations Differential equations play an important role in science, engineering, economics and many other disciplines. In this chapter we shall give a brief introduction on some of the techniques of solving differential equations. 8.1. Introduction A differential equation is an equation which contains derivatives of an unknown function. Here are a few examples of differential equations: ( i ) dy dx = x 2 , ( ii ) dP dt = kP, ( iii ) (sin y ) d 2 y dx 2 = (1- y ) dy dx + y 2 , ( iv ) d 2 u dt 2 + 5 du dt 3- 3 u = 0 , ( v ) d 3 y dx 3 2 + x dy dx- 2 y = 0 The order of a differential equation is the highest order of derivative present in the differential equation. Equations (i) and (ii) are first order differential equations, (iii) and (iv) are second order differential equations, and (v) is a third order differential equation. The degree of the equation is the algebraic degree with which the derivative of highest or- der appears in the equation. Equations (i)-(iv) are of degree 1 whereas equation (v) is of degree 2. Let us consider equation (i). It is not hard to see that y ( x ) = 1 3 x 3 + c satisfies the differ- ential equation for any constant c ∈ R , and hence it is a solution for (i). When a solution of a differential equation contains an arbitrary constant c , we call the solution a general solution of the equation. A general solution describes a family of particular solutions of the differential equation. For example, the choices c = 9 , 27 gives two particular solutions for (i), namely y 1 ( x ) = 1 3 x 3 + 9 and y 2 ( x ) = 1 3 x 3 + 27 . A differential equation frequently appears together with an initial condition (or boundary condition ) y ( a ) = b . For example, the particular solution y 1 ( x ) satisfies the initial value problem dy dx = x 2 , y (3) = 18 . 8.2. Separable Equations In this section we study a method of finding the general solution of certain kinds of first order differential equations. Consider the following differential equation: N ( y ) dy dx = M ( x ) . (8.1) Such a differential equation is said to be separable because it can also be written as the form N ( y ) dy = M ( x ) dx (8.2) in which the variables x and y are separated on opposite side of the equation. Integrating both sides of (8.2) gives Z N ( y ) dy = Z M ( x ) dx. If we write F ( y ) = R N ( y ) dy and G ( x ) = R M ( x ) dx , then the equation F ( y ) = G ( x ) + c, c ∈ R provides a general solution of (8.1). Example 8.1 Solve the following initial value problem dy dx = 4 xy, y (0) = 2 . Solution. Applying the method of separating variables gives 1 y dy = 4 x dx. Integration then yields ln | y | = 2 x 2 + c, so that | y | = e c e 2 x 2 , that is, y ( x ) = ± e c e 2 x 2 . Substituting x = 0 and y = 2 yields 2 = ± e c , which means c = ln 2 and we must have the positive sign. It follows that the desired particular solution is given by y ( x ) = 2 e 2 x 2 . Example 8.2 (Radioactivity. Exponential Decay) Denote by y ( t ) the amount of a certain...
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