MA22016 Differential Equations

Define an ordinary differential equation (ODE)

An ordinary differential equation (ODE) is an equation that relates one or more state variables to their derivatives with respect to an independent variable (t, usually representing time).

Define a parameter

A parameter is a constant that appears in the ODE but does not change with respect to the independent variable. Parameters are usually denoted by the Greek letters, eg α, β and γ.

Define the order of a derivative

The order of a derivative is the number of times the differential operator is applied.

Define the order of an ODE 

The order of an ODE is the order of highest derivative.

When is an ODE autonomous?

An ODE is autonomous if it does not explicitly depend on the independent variable t.
Otherwise, it is nonautonomous.

When is an ODE homogeneous?

An ODE is homogeneous if it does not include any terms that are either constant or depend only on the independent variable.

Otherwise, it is inhomogeneous.

Describe a linear ODE

An ODE is linear if it is defined by a linear polynomial in the dependent variable and its derivatives, ie. there are no products of the independent variable with itself or any of its derivatives. 

Otherwise, it is nonlinear.

Give the general form of a linear ODE

where a(sub i)(t) and f(t) are functions of t and x^(i) is the ith derivative of x with respect to t.

Define an initial value problem (IVP)

An initial value problem (IVP) is an ODE where the value of the dependent variable is known at some prescribed value of the independent variable, usually t(sub0) = 0

Define a continuous-time dynamical system 

A continuous time dynamical system is a system composed of one or more states that evolve over time according to one or more differential equations.

Define a single first order ODE

A single first order ODE is an equation of the form xdot = f(t,x) where f is a known function of t and x that may be linear or nonlinear.

If we wish to emphasise that f also depends on the parameter μ we write xdot = f(t, x;μ)

Define a system of first order ODEs

A system of first order ODEs is a coupled set of first order ODEs. There is one equation for each state variable in the system.

If all the equations are linear, the system is a linear system. Otherwise, it is a nonlinear system.

In general, a system of n ODEs for n state variables has the form,

xdot = f (t,x)

where xdot, f and x are all vectors

x = [x1,x2,…,xn]

f = [f(sub1)(t,x), f(sub2)(t,x),…,f(subn)(t,x)]

Define the state space

The state space is the set of all possible values of the state variables.

Define the system state

The system state is the value of the state variables at a given time t.

State Theorem 1.1 Existence and uniqueness of solutions

When is a problem well-posed?

A problem is said to be well-posed if,

1. a solution exists, at least over the time interval for which it is required

2. the solution is unique

3. the solution depends continuously on the initial condition and parameters

Define a steady state (or equilibrium) of an ODE

A steady state or equilibrium of an ODE is a state that does not change with time. So x* is a steady state, or equilibrium, of the ODE xdot = f(x) if f(x*) = 0.