Ordinary differential equation

An ordinary differential equation (ODE) relates a function and its derivatives. We usually use $y$ to denote the function and 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} to denote the variable.

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

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

Example

An example of an ODE is the following

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 }

The general solution of the above is

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) = c e^t }

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

Solving ODE

An ODE is considered solve if we can express y in terms of integrals of explicitly known functions. We don't necessarily need to solve the integral; some integrals are very very hard.

Classification

Linearity

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. As one see from Linear Algebra, for a linear problem, a general solution can be constructed from one specific solution and the homogeneous solution.

Order

The order of an ODE is the order of its highest derivative. Higher order ODEs are generally harder to solve.

Scalar/System

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 (such as predator-prey functions).

Solvable Classes

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

• Linear First Order ODE
• First Order Scalar
• Autonomous ODE

Applications

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}}

Here are some more general questions that are solved by ODEs

• Initial Value Problem: Use an initial value of a function and an ODE to generate a function.
  • Equilibrium Solution: Find the equilibrium solution of an initial value problem

Dimensional Analysis

The two sides of the equation must match in dimensions (aka. units). This also applies in ODEs.

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} }