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Ordinary Differential Equations

In: Advanced Mathematical Methods for Scientists and Engineers I

Author

Listed:
  • Carl M. Bender

    (Washington University, Department of Physics)

  • Steven A. Orszag

    (Yale University, Department of Mathematics)

Abstract

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.

Suggested Citation

  • Carl M. Bender & Steven A. Orszag, 1999. "Ordinary Differential Equations," Springer Books, in: Advanced Mathematical Methods for Scientists and Engineers I, chapter 0, pages 3-35, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-3069-2_1
    DOI: 10.1007/978-1-4757-3069-2_1
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