Differential Equations For Dummies
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The fun and easy way to understand and solve complex equations
Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
but it’s actually a little tighter than it needs to be in order to guarantee just a solution (which isn’t necessarily unique). In fact, you can show that there’s a solution — but not that it’s unique — to the nonlinear differential equation merely by proving that f is continuous. A couple of nonlinear existence and uniqueness examples In the following sections, I provide two examples that put the nonlinear exis- tence and uniqueness theorem into action. Example 1 Determine what the two
Differential Equations 61 Time to use the method of partial fractions. In this case, you get the following equation: dx v 2 - 1 dx v 2 1 d n x + dv = + - = 2 0 v _ v 2 x + i v v dv 1 + 1 or to simplify: dx v 2 dv dv x + - = 2 0 v v + 1 Ah, much better. Now you can integrate! Integrating all these terms gives you: ln| x| + ln| v 2 + 1| – ln| v| = c or: ln| x| + ln| v 2 + 1| = ln| v| + c Exponentiating both sides gives you this equation: x( v 2 + 1) = kv
using a computer. I know, using a computer feels a lot like a cop-out, but sometimes you have no other choice. Chapters 1, 2, and 3 all contain examples where I used a computer to calculate direction fields and plot some actual solutions. Now, in the following sections, I introduce one simple numerical method of solving differential equations. This method is called Euler’s method. 08_178140-ch04.qxd 4/29/08 1:02 AM Page 76 76 Part I: Focusing on First Order Differential Equations So who
this case, the solution is said to be an equilibrium solution. The series converges to that equilibrium solution. So, every term is the same: yn = f n( y 0) and: yn = f( yn) I introduce some examples in the following sections. 08_178140-ch04.qxd 4/29/08 1:05 AM Page 86 86 Part I: Focusing on First Order Differential Equations Working without a constant To better understand equilibrium solutions, say, for example, that you have savings in a bank that pays an interest rate, i, annually.
you find a solu- tion to this equation? Well, after you give it some thought, it looks a lot like this Euler equation: 2 2 d y dy x 2 - x + y = 2 0 dx dx So you can try a solution of the following form: 3 y = ! a x n + r n n = 0 Differentiating this solution gives you: dy 3 = ! a ^ r + n h x n + r -1 dx n n = 0 After differentiating again, you get: d 2 y 3 = ! a ^ r + n ^ h r + n n + r - 2 - h 2 n 1 x dx n = 0 Using series as substitutes in the original