Ordinary Differential Equation

• An Ordinary Differential Equation is defined as follows

  Let be an open set and be a continuous -differentiable function.

  Let be a function of independent variable and the derivatives of with respect to . An ODE is an equation of the form

  We call the above the explicit form.

  The implicit form involves simply setting the above to be equal to . Thus, it is an equation of the form

• A differential equation is autonomous if it does not depend on . That is, it is of the form

  • Autonomous ODEs correspond to systems that are time independent.

• A differential equation is linear if it can be written as a Linear Combination of the derivatives of . Thus

  Where is called the source term. Note that and are Continuous functions of

  • A linear ODE is homogeneous if in the above
  • A linear ODE is inhomogeneous if .
  • A linear ODE is nonlinear if the above does not hold.

• We can also have a System of ODEs. In such a case is a vector valued function of and its derivatives. It is of the following matrix form

• A function is a solution of the ODE (in implicit form) if it is -times differentiable on and

  • A general solution of an -th order ODE is a solution containing arbitrary constants of integration.
  • A particular solution is obtained by setting these constants to particular values. Typically, these constants are set to satisfy initial or boundary conditions on .
  • A singular solution is a solution that cannot be obtained by setting definite values to constants in the general solution.

Topics §

• ODE - Notation Guide

• Differential 1-Form

• Integral Curve

• Linear ODE

• Phase Flow

• Diffeomorphism

  • Solving Differential Equations with Diffeomorphisms

• Tangent Space

Links §

• Ordinary Differential Equations by Arnold