Differential Equations For Dummies book cover

Differential Equations For Dummies

Author:
Steven Holzner
Published: June 3, 2008

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.

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.

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.

Articles From The Book

4 results

Calculus Articles

Distinguishing among Linear, Separable, and Exact 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.) For example, this is a linear differential equation because it contains only derivatives raised to the first power:

Separable differential equations can be written so that all terms in x and all terms in y appear on opposite sides of the equation. Here's an example:

which can be written like this with a little reshuffling:

Exact differential equations are those where you can find a function whose partial derivatives correspond to the terms in a given differential equation.

Calculus Articles

Using the Method of Undetermined Coefficients

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). Because g(x) is only a function of x, you can often guess the form of yp(x), up to arbitrary coefficients, and then solve for those coefficients by plugging yp(x) into the differential equation.

This method works because you're dealing only with g(x), and the form of g(x) can often tell you what a particular solution looks like.

Calculus Articles

Classifying Differential Equations by Order

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!). Differential equations of all orders can use the y' notation, like this: