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Unformatted text preview: 5 Linear First-Order Equations “Linear” first-order differential equations make up another important class of differential equa- tions that commonly arise in applications and are relatively easy to solve (in theory). As with the notion of ‘separability’, the idea of ‘linearity’ for first-order equations can be viewed as a simple generalization of the notion of direct integrability, and a relatively straightforward (though, per- haps, not so intuitively obvious) method will allow us to put any first-order linear equation into a form that can be relatively easily integrated. We will derive this method in a short while (after, of course, describing just what it means for a first-order equation to be “linear”). By the way, the criteria given here for a differential equation being linear will be extended later to higher-order differential equations, and a rather extensive theory will be developed to handle linear differential equations of any order. That theory is not needed here; in fact, it would be of very limited value. And, to be honest, the basic technics we’ll develop in this chapter are only of limited use when it comes to solving higher-order linear equations. However, these basic technics involve an “integrating factor”, which is something we’ll be able to generalize a little bit later (in chapter 7) to help solve much more general first-order differential equations. 5.1 Basic Notions Definitions A first-order differential equation is said to be linear if and only if it can be written as dy dx = f ( x ) − p ( x ) y (5.1) or, equivalently, as dy dx + p ( x ) y = f ( x ) (5.2) where p ( x ) and f ( x ) are known functions of x only. Equation (5.2) is normally considered to be the “standard” form for first-order linear equa- tions. Note that the only appearance of y in a linear equation (other than in the derivative) is in a term where y alone is multiplied by some formula of x . If there is any other functions of y appearing in the equation after you’ve isolated the derivative, then the equation is not linear. 103 104 Linear First-Order Equations ! ◮ Example 5.1: Consider the differential equation x dy dx + 4 y − x 3 = . Solving for the derivative, we get dy dx = x 3 − 4 y x = x 2 − 4 x y , which is dy dx = f ( x ) − p ( x ) y with p ( x ) = 4 x and f ( x ) = x 2 . Adding 4 / x · y to both sides, we then get the equation in standard form, dy dx + 4 x y = x 2 , On the other hand dy dx + 4 x y 2 = x 2 is not linear because of the y 2 . In testing whether a given first-order differential equation is linear, it does not matter whether you attempt to rewrite the equation as dy dx = f ( x ) − p ( x ) y or as dy dx + p ( x ) y = f ( x ) ....
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