Ordinary Differential Equations

A first order ordinary differential equation or ODE describes the relationship between

a function and its derivative.

They follow the general form of the equation you see on

the screen

where x of t is the dependent variable

t is the independent variable often

representing time

dx by dt is the derivative of x with respect to t representing the rate of change

of x.

f is the function that defines how this rate of change of x depends on t and also y

parameters like theta.

In other words, a first order ODE specifies how a system changes over time,

where the rate of change of the system state x is determined by the current state, time,

and other influencing parameters. x of t is a function that depends on t, of course,

but how this function value changes is also dependent on time

and this relationship is

defined by f.

Let's dive into an example.

Many real world phenomena follow this ODE pattern.

For example, consider a bank account where the amount of money x grows over time due to a

compound interest. The rate of change of amount of money in the account is proportional to the

current amount of money.

This can be modeled by the ODE that you see on the screen.

Here x of t

is the amount of money at any time t

dx by dt defines the rate of change of the amount of money

with respect to time.

r is the interest rate, which is a constant parameter for in our case theta.

In this example

the function f from the general form is simply r times x of t

indicating that the rate of change of money is directly proportional to the current amount of

money and also the constant parameter r is directly linked to the constant parameter r.

The parameter theta is in this case r.

So here x of t, the amount of money is dependent on time.

The longer it's been, the more money is there.

But how much the money changes with time is defined

by f.

But how do you solve this equation?

How do we know x of t at say a certain time t1?

That leads us to the formulation of the initial value problem.

An initial value problem is you

know the initial value of the state x of t0 is equal to

for example

x0.

How you would like to find out x of t1 at some time t1?

It's a very important formulation that

has applications in many physical phenomena

especially in time series like climate change.

The solution