Differential Equation

• Differential Equations can be used to study processes that are deterministic, finite-dimensional, and differentiable.
• The fundamental problem of the theory of differential equations is to determine or study the motion of some system using the phase velocity vector field.
• Differential equations arise when the dynamics of a system is described by changes in state rather than explicit state values. Analysis proceeds by transforming the problem into an analogous geometric one in a vector field

Ordinary Differential Equations §

• An integral curve is a line that, at each point, is tangent to a vector field

  A necessary and sufficient condition for the graph of function to be an integral curve is that the following relation hold for interval [^note_1]

  The integral line lies in the direct product of the time axis and phase space (called the extended phase space)

• A function is a solution of the differential equation if

  satisfies the initial condition if

Notation §

• - time
• - element in phase space
• - extended phase space
• - interval in the time axis
• - region in phase space

Topics §

• Differential 1-Form

Partial Differential Equations §

Links §

• Paul’s Online Notes - Differential Equations

• 3B1B - Differential Equations

• Ordinary Differential Equations by Arnold

• Differential Calculus

• Integral Calculus