Ordinary differential equation

An ordinary differential equation (ODE) relates a function and its derivatives. We usually use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} to denote the function and $t$ to denote the variable.

Ordinary means that the equation has one variable, as opposed to partial differential.

There is no general solution to ODEs. We separate them by classes and solve them individually.

Example

An example of an ODE is the following

$y'=y$

The general solution of the above is

$y(t)=ce^{t}$

Notably, the solution is homogeneous, meaning that $0$ is a solution. This will probably be covered later.

To get a unique solution, we need to apply additional conditions, such as specifying a particular value

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} y' = y \\ y(0) = y_0 \end{cases} }

This is called an initial value problem, in which a function is generated from an initial value with another equation.

Usage

Since the derivative can be described as the rate of change, and the function itself is the state, ODEs arises as mathematical models of systems whose rate of change depends on the state of the system.

The following are brief descriptions of some applications of ODEs.

1. Radioactive decay, where the function is the (large) number of atoms.
   • Atoms decay at an average constant rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}
   • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dN}{dt} = -rN}
2. Object falling under gravity, where the function is the velocity of the object
   • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dv}{dt} = g - \frac{\gamma v}{m}}

Dimensions/Units

The two sides of the equation must match in dimensions (aka. units).

Consider radioactive decay.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} \frac{dN}{dt} = -rN \\ N(0) = N_0 \end{cases} }

The solution comes to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(t) = N_0 e^{-rt} }

We use time constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} to get a sense of how fast it is decaying. Its units is time.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = \frac{1}{r} }

Equilibrium Solution

Consider an object falling under gravity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} \frac{dv}{dt} = g - \lambda v \\ v(0) = v_0 \end{cases} }

We sometimes want the equilibrium solution

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t) = v_* }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dv}{dt} = 0 = g - \lambda v_* }

Doing some math, we can eventually get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t) = v_* + (v_0 - v_*) e^{-\lambda t} }

Classification

An ODE is linear if all terms are proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y, y', y''. \ldots} or are given functions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} . This distinction is especially useful since linear combination can be used to construct solutions.

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

In a scalar, there is only one unknown function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t)} . In a system, there are several, and you have to solve them simultaneously.

Here is a list of ODEs we study, from simple to complex:

• Linear First Order ODE