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Unformatted text preview: 4 Separable First-Order Equations As we will see below, the notion of a differential equation being separable is a natural general- ization of the notion of a first-order differential equation being directly integrable. Whats more, a fairly natural modification of the method for solving directly integrable first-order equations gives us the basic approach to solving separable differential equations. However, it cannot be said that the theory of separable equations is just a trivial extension of the theory of directly integrable equations. Certain issues can arise that do not arise in solving directly integrable equa- tions. Some of these issues are pertinent to even more general classes of first-order differential equations than those that are just separable, and may play a role later on in this text. In this chapter we will, of course, learn how to identify and solve separable first-order differential equations. We will also see what sort of issues can arise, examine those issues, and discuss some ways to deal with them. Since many of these issues involve graphing, we will also draw a bunch of pictures. 4.1 Basic Notions Separability A first-order differential equation is said to be separable if, after solving it for the derivative, dy dx = F ( x , y ) , the right-hand side can then be factored as a formula of just x times a formula of just y , F ( x , y ) = f ( x ) g ( y ) . If this factoring is not possible, the equation is not separable. More concisely, a first-order differential equation is separable if and only if it can be written as dy dx = f ( x ) g ( y ) (4.1) where f and g are known functions. ! Example 4.1: Consider the differential equation dy dx x 2 y 2 = x 2 . (4.2) 73 74 Separable First-Order Equations Solving for the derivative (by adding x 2 y 2 to both sides), dy dx = x 2 + x 2 y 2 , and then factoring out the x 2 on the right-hand side gives dy dx = x 2 ( 1 + y 2 ) , which is in form dy dx = f ( x ) g ( y ) with f ( x ) = x 2 bracehtipupleftbracehtipdownrightbracehtipdownleftbracehtipupright no y s and g ( y ) = ( 1 + y 2 ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright no x s . So equation (4.2) is a separable differential equation. ! Example 4.2: On the other hand, consider dy dx x 2 y 2 = 4 . (4.3) Solving for the derivative here yields dy dx = x 2 y 2 + 4 . The right-hand side of this clearly cannot be factored into a function of just x times a function of just y . Thus, equation (4.3) is not separable. We should (briefly) note that any directly integrable first-order differential equation dy dx = f ( x ) can be viewed as also being the separable equation dy dx = f ( x ) g ( y ) with g ( y ) being the constant 1 . Likewise, a first-order autonomous differential equation dy dx = g ( y ) can also be viewed as being separable, this time with f ( x ) being 1 . Thus, both directly integrable and autonomous differential equations are all special cases of separable differential equations. Basic Notions...
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