### Features

Contains solutions to more than 6200 ODEs-more than any other reference availableOffers introductory material describing key concepts, theorems, and methodsPays special attention to equations dependent on arbitrary functionsCovers Abel, Emden-Fowler, and Painlevé equationsIncludes supplemental material on the gamma, beta, Bessel, Legendre, Weierstrass, hypergeometric, and other special functions
### Summary

Exact solutions of differential equations continue to play an important role in the understanding of many phenomena and processes throughout the natural sciences in that they can verify the correctness of or estimate errors in solutions reached by numerical, asymptotic, and approximate analytical methods. The new edition of this bestselling handbook now contains the exact solutions to more than 6200 ordinary differential equations. The authors have made significant enhancements to this edition, including:

An introductory chapter that describes exact, asymptotic, and approximate analytical methods for solving ordinary differential equations

The addition of solutions to more than 1200 nonlinear equations

An improved format that allows for an expanded table of contents that makes locating equations of interest more quickly and easily

Expansion of the supplement on special functions

This handbook's focus on equations encountered in applications and on equations that appear simple but prove particularly difficult to integrate make it an indispensable addition to the arsenals of mathematicians, scientists, and engineers alike.

### Table of Contents

INTRODUCTION: SOME DEFINITIONS, FORMULAS, METHODS, AND TRANSFORMATIONS

First-Order Differential Equations

Second-Order Differential Equations

Second-Order Nonlinear Differential Equations

Linear Equations o Arbitrary Order

Nonlinear Equations of an Arbitrary Order

Group Methods and Discrete Group Analysis

FIRST ORDER DIFFERENTIAL EQUATIONS

Simplest Equations with Arbitrary Functions Integrable in a Closed Form

Riccati Equation

Abel Equations of the Second Kind

Equations Containing Polynomial Functions of y

Equations of the Form f(x,y)y'x = g(x,y) Containing Arbitrary Parameters

Equations of the Form F(x,y,y'x) = 0 Containing Arbitrary Parameters

Equations of the Form F(x,y)y'x = g(x,y) Containing Arbitrary Functions

Equations of the Form F(x,y,y'x) = 0 Containing Arbitrary Functions

SECOND ORDER DIFFERENTIAL EQUATIONS

Linear Equations

Autonomous Equations

Emden-Fowler Equation

Equations of the Form y"xx = A1xn1ym1 + A2xn2ym2

Generalized Emden-Fowler Equation

Equations of the Form y"xx = A1xn1ym1(y'x)l1 + A2xn2ym2(y'x)l2

Equations of the Form y"xx = f(x)g(y)h(y'x)

Some Nonlinear Equations with Arbitrary Parameters

Equations Containing Arbitrary Functions

THIRD ORDER DIFFERENTIAL EQUATIONS

Linear Equations

Equations of the Form y'"xxx = Axαyß(y'x) γ(y"xx)δ

Equations of the Form y'"xxx = f(y)g(y'x)h(y"xx)

Some Nonlinear Equations with Arbitrary Parameters

Nonlinear Equations Containing Arbitrary Functions

FOURTH ORDER DIFFERENTIAL EQUATIONS

Linear Equations

Nonlinear Equations

HIGHER ORDER DIFFERENTIAL EQUATIONS

Linear Equations

Nonlinear Equations

SUPPLEMENTS

Elementary Functions and Their Properties

Special Functions and Their Properties

Table of Indefinite Integrals

REFERENCES

INDEX

### Reviews

"…I have been reading the Polyanin books "Handbook of Linear Partial Differential Equations for Engineers and Scientists" and "Handbook of Exact Solutions for Ordinary Differential Equations". I think these books are extraordinary, and are destined to become classics … CRC Press has provided an invaluable service to science and engineering by publishing these books."

- William Schiesser, Lehigh University

"In summary, this is a very useful and easy-to-read handbook for anyone involved in ordinary differential equations..."

- SIAM Review, 1997, on the first edition

"This best-selling handbook ... is now in its second edition and has been substantially updated and improved ... book is highly recommended for mathematicians, scientists, and

engineers. It would be a useful edition to academic sci-tech libraries, including those that already have the first edition."

- Holly Flynn in E-STREAM