What is an ODE (ODEs)

ODE stands for Ordinary Differential Equation.

An ordinary differential equation is a mathematical equation that contains a function and its derivatives with respect to a single independent variable (usually time, denoted as t, or position, denoted as x).

Key Characteristics of ODEs:

• Single independent variable: Unlike partial differential equations (PDEs) which involve multiple independent variables
• Contains derivatives: The equation includes derivatives like $\frac{dx}{dt}$, $\frac{d^{2}x}{d{t}^2}$, etc.
• Describes dynamic systems: Often used to model how quantities change over time

Your Example Explained:

The equation $\frac{dx}{dt}=t$ is an ODE where:

• $x$ is the dependent variable (function we’re solving for)
• $t$ is the independent variable (time)
• The equation shows how $x$ changes with respect to time

This is a time-dependent or non-autonomous ODE because the right side explicitly contains the independent variable $t$.

Autonomous vs Non-Autonomous ODEs:

• Autonomous: $\frac{dx}{dt}=f(x)$ - doesn’t explicitly depend on time
• Non-autonomous: $\frac{dx}{dt}=f(x,t)$ - explicitly depends on time (like your example)

When you mention including $t$ as another state variable, you’re referring to a common technique where a non-autonomous system can be converted to an autonomous one by treating time as an additional state variable with $\frac{dt}{dt}=1$.

ODEs are fundamental in physics, engineering, biology, and economics for modeling dynamic systems and understanding how quantities evolve over time.