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Published:
June 3, 2008

Differential Equations For Dummies

Overview

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.

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About The Author

Steven Holzner is an award-winning author of science, math, and technical books. He got his training in differential equations at MIT and at Cornell University, where he got his PhD. He has been on the faculty at both MIT and Cornell University, and has written such bestsellers as Physics For Dummies and Physics Workbook For Dummies.

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differential equations for dummies

CHEAT SHEET

To confidently solve differential equations, you need to understand how the equations are classified by order, how to distinguish between linear, separable, and exact equations, and how to identify homogenous and nonhomogeneous differential equations.Learn the method of undetermined coefficients to work out nonhomogeneous differential equations.

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The most common classification of differential equations is based on order. The order of a differential equation simply is the order of its highest derivative. You can have first-, second-, and higher-order differential equations. First-order differential equations involve derivatives of the first order, such as in this example: Second-order differential equations involve derivatives of the second order, such as in these examples: Higher-order differential equations are those involving derivatives higher than the second order (big surprise on that clever name!
In order to identify a nonhomogeneous differential equation, you first need to know what a homogeneous differential equation looks like. You also often need to solve one before you can solve the other. Homogeneous differential equations involve only derivatives of y and terms involving y, and they're set to 0, as in this equation: Nonhomogeneous differential equations are the same as homogeneous differential equations, except they can have terms involving only x (and constants) on the right side, as in this equation: You also can write nonhomogeneous differential equations in this format: y'' + p(x)y' + q(x)y = g(x).
To confidently solve differential equations, you need to understand how the equations are classified by order, how to distinguish between linear, separable, and exact equations, and how to identify homogenous and nonhomogeneous differential equations.Learn the method of undetermined coefficients to work out nonhomogeneous differential equations.
You can distinguish among linear, separable, and exact differential equations if you know what to look for. Keep in mind that you may need to reshuffle an equation to identify it. Linear differential equations involve only derivatives of y and terms of y to the first power, not raised to any higher power. (Note: This is the power the derivative is raised to, not the order of the derivative.
If you need to find particular solutions to nonhomogeneous differential equations, then you can start with the method of undetermined coefficients. Suppose you face the following nonhomogeneous differential equation: The method of undetermined coefficients notes that when you find a candidate solution, y, and plug it into the left-hand side of the equation, you end up with g(x).
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