Ordinary differential equations

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• ConceptOrdinary differential equation

  An ordinary differential equation is a differential equation for an unknown function of one variable. It relates the function, its derivatives and the variable. See the definition.

• ConceptOrder of a differential equation

  The order of a differential equation is the highest order of a derivative of the unknown function in it. See the definition.

• ConceptInitial value problem

  An initial value problem is an ordinary differential equation together with a condition that fixes the value of the unknown function at one point. A solution must satisfy both the equation and the condition. See the definition.

• ConceptAutonomous system

  An autonomous system is a system of ordinary differential equations $x' = f(x)$ whose right side does not depend explicitly on the independent variable. When the independent variable is time, such a system is also called time-invariant. See the definition.

• ConceptSolution of an ordinary differential equation

  A solution of an ordinary differential equation is a function on an interval that satisfies the equation at each point. For a first-order system given by a vector field, the solutions are the integral curves. These are the curves whose velocity is the value of the vector field at each point. See the definition.

• DefinitionSlope field

  Let $\Omega$ be an open subset of $\mathbb{R}^2$ and let $f : \Omega \to \mathbb{R}$ be a function. The slope field of the first-order equation $y' = f(x, y)$ is the map that assigns to each point $(x, y) \in \Omega$ the line through $(x, y)$ with slope $f(x, y)$. In a picture, one draws a short segment of this line at each point of a grid in $\Omega$.