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Unformatted text preview: 3 Some Basics about First-Order Equations For the next few chapters, our attention will be focused on first-order differential equations. We will discover that these equations can often be solved using methods developed directly from the tools of elementary calculus. And even when these equations cannot be explicitly solved, we will still be able to use fundamental concepts from elementary calculus to obtain good approximations to the desired solutions. But first, let us discuss a few basic ideas that will be relevant throughout our discussion of first-order differential equations. 3.1 Algebraically Solving for the Derivative Here are some of the first-order differential equations that we have seen or will see in the next few chapters: x 2 dy dx 4 x = 6 , dy dx x 2 y 2 = x 2 , dy dx + 4 xy = 2 xy 2 , and x dy dx + 4 y = x 3 . One thing we can do with each of these equations is to algebraically solve for the derivative. Doing this with the first equation: x 2 dy dx 4 x = 6 equal1 x 2 dy dx = 6 + 4 x equal1 dy dx = 4 x + 6 x 2 . 43 44 Some Basics about First-Order Equations For the second equation: dy dx x 2 y 2 = x 2 equal1 dy dx = x 2 + x 2 y 2 . Solving for the derivative is often a good first step towards solving a first-order differential equation. For example, the first equation above is directly integrable solving for the derivative yielded dy dx = 4 x + 6 x 2 , and y ( x ) can now be found by simply integrating both sides with respect to x . Even when the equation is not directly integrable and we get dy dx = a formula of both x and y , as in our second equation above, dy dx = x 2 + x 2 y 2 that formula on the right can still give us useful information about the possible solutions and can help us determine which method is appropriate for obtaining the general solution. Observe, for example, that the right-hand side of the last equation can be factored into a formula of x and a formula of y , dy dx = x 2 ( 1 + y 2 ) . In the next chapter, we will find that this means the equation is separable and can be solved by a procedure developed for just such equations. For convenience, let us say that a first-order differential equation is in derivative formula form if it is written as dy dx = F ( x , y ) (3.1) where F ( x , y ) is some (known) formula of x and/or y . Remember, to convert a given first- order differential equation to derivative form, simply use a little algebra to solve the differential equation for the derivative. ? Exercise 3.1: Verify that the derivative formula forms of dy dx + 4 y = 3 y 3 and x dy dx + 4 xy = 2 y 2 are dy dx = 3 y 3 4 y and dy dx = 2 y 2 4 xy x , respectively....
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