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Unformatted text preview: 12 Higher-Order Linear Equations: Introduction and Basic Theory We have just seen that some higher-order differential equations can be solved using methods for first-order equations after applying the substitution v = dy / dx . Unfortunately, this approach has its limitations. Moreover, as we will later see, many of those differential equations that can be so solved can also be solved much more easily using the theory and methods that will be developed in the next few chapters. This theory and methodology apply to the class of linear differential equations. This is a rather large class that includes a great many differential equations arising in applications. In fact, so important is this class of equations and so extensive is the theory for dealing with these equations, that we will not seriously consider higher-order nonlinear differential equations (excluding those in the previous chapter) for many, many chapters. 12.1 Basic Terminology Recall that a first-order differential equation is said to be linear if and only it can be written as dy dx + py = f (12.1) where p = p ( x ) and f = f ( x ) are known functions. Observe that this is the same as saying that a first-order differential equation is linear if and only if it can be written as a dy dx + by = g (12.2) where a , b , and g are known functions of x . After all, the first equation is equation (12.2) with a = 1 , b = p and f = g , and any equation in the form of equation (12.2) can be converted to one looking like equation (12.1) by simply dividing through by a (so p = b / a and f = g / a ). Higher order analogs of either equation (12.1) or equation (12.2) can be used to define when a higher-order differential equation is linear. We will find it slightly more convenient to use analogs of equation (12.2) (which was the reason for the above observations). Second- and third-order linear equations will first be described so you can start seeing the pattern. Then the general definition will be given. For convenience (and because there are only so many letters in the alphabet), we may start denoting different functions with subscripts. 259 260 Higher-Order Linear Equations: Definitions and Some Basic Theory A second-order differential equation is said to be linear if and only if it can be written as a d 2 y dx 2 + a 1 dy dx + a 2 y = g (12.3) where a , a 1 , a 2 , and g are known functions of x . (In practice, generic second-order differ- ential equations are often denoted by a d 2 y dx 2 + b dy dx + cy = g , instead.) For example, d 2 y dx 2 + x 2 dy dx 6 x 4 y = x + 1 and 3 d 2 y dx 2 + 8 dy dx 6 y = are second-order linear differential equations, while d 2 y dx 2 + y 2 dy dx = x + 1 and d 2 y dx 2 = parenleftbigg dy dx parenrightbigg 2 are not....
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