{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Extending_First_Order

# Extending_First_Order - 11 Higher-Order Equations Extending...

This preview shows pages 1–3. Sign up to view the full content.

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 = 0 , 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 solvinghigher-orderequationsmoreofachallenge. Thisdoesnotmeanthatthoseideasdeveloped 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 v using methods discussed in previous chapters, and y can then be obtained from v
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}