👉 Ordinary differential equations (ODEs) are mathematical equations that describe how quantities change over time or space, given initial conditions. They are fundamental in modeling dynamic systems across various fields such as physics, biology, economics, and engineering. An ODE typically involves a function of one or more independent variables (often time) and its derivatives, expressing the rate of change of the function with respect to those variables. For example, in physics, Newton's second law can be written as an ODE describing the motion of a particle under the influence of forces. Solving ODEs involves finding functions that satisfy the given equation and initial conditions, which provide specific starting points for the solution. Techniques to solve ODEs range from analytical methods like separation of variables and integrating factors to numerical methods for more complex cases, allowing us to predict and understand the behavior of dynamic systems over time.
