Ordinary Differential Equations

  • Carl M. Bender
  • Steven A. Orszag


An nth-order differential equation has the form
$$ {y^{\left( n \right)}}\left( x \right) = F\left[ {x,y\left( x \right),y'\left( x \right), \ldots ,{y^{\left( {n - 1} \right)}}\left( x \right)} \right], $$
where y (k)= d k y/dx k . Equation (1.1.1) is a linear differential equation if F is a linear function of y and its derivatives (the explicit x dependence of F is still arbitrary). If (1.1.1) is linear, then the general solution y(x) depends on n independent parameters called constants of integration; all solutions of a linear differential equation may be obtained by proper choice of these constants. If (1.1.1) is a nonlinear differential equation, then it also has a general solution which contains n constants of integration. However, there sometimes exist special additional solutions of nonlinear differential equations that cannot be obtained from the general solution for any choice of the integration constants. We omit a rigorous discussion of these fundamental properties of differential equations but illustrate them in the next three examples.


General Solution Riccati Equation Nonlinear Differential Equation Linear Differential Equation Bernoulli Equation 
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Chapter 1 Some good general texts on differential equations

  1. 1.
    Birkhoff, G., and Rota, G. C, Ordinary Differential Equations, Ginn and Company, Boston, 1962.zbMATHGoogle Scholar
  2. 2.
    Boyce, W. E., and DiPrima, R. C, Elementary Differential Equations, John Wiley and Sons, Inc., New York, 1969.zbMATHGoogle Scholar
  3. 3.
    Carrier, G., and Pearson, C. E., Ordinary Differential Equations, Blaisdell Publishing Company, Waltham, Mass., 1968.zbMATHGoogle Scholar

Chapter 1 Advanced texts on differential equations

  1. 4.
    Coddington, E. A., and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York, 1955.Google Scholar
  2. 5.
    Ince, E. L., Ordinary Differential Equations, Dover Publications, Inc., New York, 1956.Google Scholar

Chapter 1 Green’s function, δ functions, transform methods, and eigenfunction expansions are discussed by

  1. 6.
    Courant, R., and Hubert, D., Methods of Mathematical Physics, vol. 1, Interscience Publishers, New York, 1953.Google Scholar
  2. 7.
    Jeffreys, H., and Jeffreys, B. S., Methods of Mathematical Physics, 3d ed., Cambridge University Press, London, 1956.zbMATHGoogle Scholar
  3. 8.
    Lighthill, M. J., Introduction to Fourier Analysis and Generalized Functions, Cambridge University Press, London, 1958.CrossRefGoogle Scholar
  4. 9.
    Morse, P. M., and Feshbach, H., Methods of Theoretical Physics, pt. I, McGraw-Hill Book Company, Inc., New York, 1953.zbMATHGoogle Scholar

Chapter 1 Useful reference works on the special functions of mathematical physics and applied mathematics

  1. 10.
    Abramowitz, M., and Stegan, I. A., Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1964.zbMATHGoogle Scholar
  2. 11.
    Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricorni, F. G., Higher Transcedental Functions, 3 vols, McGraw-Hill Book Company, Inc., New York, 1953.Google Scholar
  3. 12.
    Whittaker, E. T., and Watson, G. N., A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1927.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Carl M. Bender
    • 1
  • Steven A. Orszag
    • 2
  1. 1.Department of PhysicsWashington UniversitySt. LouisUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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