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# Ordinary Differential Equations

Module by: Aurèle Balavoine. E-mail the author

Summary: Short introduction to the theory of Ordinary Differential Equations, their solutions and the concept of a phase portrait.

Read more about initial value problem, stability of equilibrium and Lyapunov direct method.

## Ordinary Differential Equation

The evolution of the state of many real systems, such as a planet or a stock market, can be described through a set of equations called differential equations. They involve the first derivative (with respect to time in general) of a variable and sometimes higher order derivatives. For instance, in mechanics, the famous Newton's second law of motion yields a differential equation that involves the second order derivative of a variable describing the position of an object.

To set notation, let tt be a positive scalar and xx in RNRN a vector of dimension NN. An Ordinary Differential Equation (ODE) is a relation of the form:

x ˙ ( t ) = f ( t , x ( t ) ) . x ˙ ( t ) = f ( t , x ( t ) ) .
(1)

The function f:DRNf:DRN is continuous on DD, a subset of R×RNR×RN, and x˙=dx/dtx˙=dx/dt.

## Higher order and complex ODEs

Note that the definition in Equation 1 encompasses nthnth order differential equations (i.e., equations that contains derivatives up to order nn), as well as complex valued and vector-valued differential equations. Indeed, we can rewrite any nthnth order differential equation of the form:

u ( n ) ( t ) = F ( t , u ( t ) , u ' ( t ) , ... , u ( n - 1 ) ( t ) ) u ( n ) ( t ) = F ( t , u ( t ) , u ' ( t ) , ... , u ( n - 1 ) ( t ) )
(2)

where u(i)=diu/dtiu(i)=diu/dti, in the form Equation 1 by letting:

x ( x 1 , x 2 , ... , x n ) = ( u , u ( 1 ) , ... , u ( n - 1 ) ) f = ( x 2 , ... , x n , F ) . x ( x 1 , x 2 , ... , x n ) = ( u , u ( 1 ) , ... , u ( n - 1 ) ) f = ( x 2 , ... , x n , F ) .
(3)

Then, we have:

x ˙ ( t ) = d d t u ( t ) u ( 1 ) ( t ) u ( n - 1 ) ( t ) = u ( 1 ) ( t ) u ( 2 ) ( t ) u ( n ) ( t ) = x 2 ( t ) x n ( t ) F ( t , x ( t ) ) = f ( t , x ( t ) ) . x ˙ ( t ) = d d t u ( t ) u ( 1 ) ( t ) u ( n - 1 ) ( t ) = u ( 1 ) ( t ) u ( 2 ) ( t ) u ( n ) ( t ) = x 2 ( t ) x n ( t ) F ( t , x ( t ) ) = f ( t , x ( t ) ) .
(4)

Finally, note that the solution to a complex-valued differential equation can be obtained by taking the real and imaginary parts and solving a system of the form Equation 1.

## Solution

A function xx is a solution of Equation 1 on an interval IRIR if:

• it is defined and continuously differentiable on II and
• for all tt in II, we have (t,x(t))D(t,x(t))D and xx satisfies Equation 1 on II.

The function ff is sometimes referred to as vector field on DD. If the vector field ff is independent of tt and only depends on xx, then the differential equation is called autonomous.

## Example

Consider the following differential equation for xR2xR2:

x ˙ ( t ) = 2 0 0 - 1 / 2 x ( t ) , x ˙ ( t ) = 2 0 0 - 1 / 2 x ( t ) ,
(5)

which we can rewrite as:

x ˙ 1 ( t ) = 2 x 1 ( t ) x ˙ 2 ( t ) = - 1 / 2 x 2 ( t ) . x ˙ 1 ( t ) = 2 x 1 ( t ) x ˙ 2 ( t ) = - 1 / 2 x 2 ( t ) .
(6)

Then, we can represent the vector field f:(x1,x2)(2x1,-1/2x2)f:(x1,x2)(2x1,-1/2x2) at each point (x1,x2)(x1,x2) in the plane by the corresponding vector f(x1,x2)f(x1,x2) as in Figure 1. The vectors are in the direction of f(x1,x2)f(x1,x2) and their length is proportional to the magnitude of the vector. Figure 1 is called a phase portrait. Since the vector field does not depend on tt directly, this differential equation is autonomous.

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