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Unformatted text preview: 11 Higher-Order Equations: Extending First-Order Concepts Let us switch our attention from first-order differential equations to differential equations of order two or higher. Our main interest will be with second-order differential equations, both because it is natural to look at second-order equations after studying first-order equations, and because second-order equations arise in applications much more often than do third-, or fourth- or eighty-third-order equations. Some examples of second-order differential equations are 1 y ′′ + y = , y ′′ + 2 xy ′ − 5 sin ( x ) y = 30 e 3 x , and ( y + 1 ) y ′′ = ( y ′ ) 2 . Still, even higher order differential equations, such as 8 y ′′′ + 4 y ′′ + 3 y ′ − 83 y = 2 e 4 x , x 3 y ( iv ) + 6 x 2 y ′′ + 3 xy ′ − 83 sin ( x ) y = 2 e 4 x , and y ( 83 ) + 2 y 3 y ( 53 ) − x 2 y ′′ = 18 , can arise in applications, at least on occasion. Fortunately, many of the ideas used in solving these are straightforward extensions of those used to solve second-order equations. We will make use of this fact extensively in the following chapters. Unfortunately, though, the methods we developed to solve first-order differential equations are of limited direct use in solving higher-order equations. Remember, most of those methods were based on integrating the differential equation after rearranging it into a form that could be legitimately integrated. This rarely is possible with higher-order equations, and that makes solving higher-order equations more of a challenge. This does not mean that those ideas developed in previous chapters are useless in solving higher-order equations, only that their use will tend to be subtle rather than obvious. Still, there are higher-order differential equations that, after the application of a simple substitution, can be treated and solved as first-order equations. While our knowledge of first- order equations is still fresh, let us consider some of the more important situations in which this is 1 For notational brevity, we will start using the ‘prime’ notation for derivatives a bit more. It is still recommended, however, that you use the ‘ d / dx ’ notation when finding solutions just to help keep track of the variables involved. 241 242 Higher-Order Equations: Extending First-Order Concepts possible. We will also take a quick look at how the basic ideas regarding first-order initial-value problems extend to higher-order initial-value problems. And finally, to cap off this chapter, we will briefly discuss the higher-order extensions of the existence and uniqueness theorems from section 3.3. 11.1 Treating Some Second-Order Equations as First-Order Suppose we have a second-order differential equation (with y being the yet unknown function and x being the variable). With luck, it is possible to convert the given equation to a first-order differential equation for another function v via the substitution v = y ′ . With a little more luck, that first-order equation can then be solved for...
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