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**Unformatted text preview: **2 Integration and Differential Equations Often, when attempting to solve a differential equation, we are naturally led to computing one or more integrals — after all, integration is the inverse of differentiation. Indeed, we have already solved one simple second-order differential equation by repeated integration (the one arising in the simplest falling object model, starting on page 10). Let us now briefly consider the general case where integration is immediately applicable, and also consider some practical aspects of using both the indefinite integral and the definite integral. 2.1 Directly-Integrable Equations We will say that a given first-order differential equation is directly integrable if (and only if) it can be (re)written as dy dx = f ( x ) (2.1) where f ( x ) is some known function of just x (no y ’s ). More generally, any N th-order differ- ential equation will be said to be directly integrable if and only if it can be (re)written as d N y dx N = f ( x ) (2.1 ′ ) where, again, f ( x ) is some known function of just x (no y ’s or derivatives of y ). ! ◮ Example 2.1: Consider the equation x 2 dy dx − 4 x = 6 . (2.2) Solving this equation for the derivative: x 2 dy dx = 4 x + 6 equal1⇒ dy dx = 4 x + 6 x 2 . Since the right-hand side of the last equation depends only on x , we do have dy dx = f ( x ) parenleftbigg with f ( x ) = 4 x + 6 x 2 parenrightbigg . 23 24 Integration and Differential Equations So equation (2.2) is directly integrable. ! ◮ Example 2.2: Consider the equation x 2 dy dx − 4 xy = 6 . (2.3) Solving this equation for the derivative: x 2 dy dx = 4 xy + 6 equal1⇒ dy dx = 4 xy + 6 x 2 . Here, the right-hand side of the last equation depends on both x and y , not just x . So equation (2.3) is not directly integrable. Solving a directly-integrable equation is easy: First solve for the derivative to get the equation into form (2.1) or (2.1 ′ ), then integrate both sides as many times as needed to eliminate the derivatives, and, finally, do whatever simplification seems appropriate. ! ◮ Example 2.3: Again, consider x 2 dy dx − 4 x = 6 . (2.4) In example 2.1, we saw that it is directly integrable and can be rewritten as dy dx = 4 x + 6 x 2 . Integrating both sides of this equation with respect to x (and doing a little algebra): integraldisplay dy dx dx = integraldisplay 4 x + 6 x 2 dx (2.5) equal1⇒ y ( x ) + c 1 = integraldisplay bracketleftbigg 4 x + 6 x 2 bracketrightbigg dx = 4 integraldisplay x − 1 dx + 6 integraldisplay x − 2 dx = 4 ln | x | + c 2 − 6 x − 1 + c 3 where c 1 , c 2 , and c 3 are arbitrary constants. Rearranging things slightly and letting c = c 2 + c 3 − c 1 , this last equation simplifies to y ( x ) = 4 ln | x | − 6 x − 1 + c ....

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