A differential equation is an equation that relates an unknown function to its derivatives, and its solution is a function, not a number. These equations are used to describe how quantities change, making them vital in fields like physics, engineering, biology, and economics. The main types are ordinary differential equations (ODEs), which involve ordinary derivatives, and partial differential equations (PDEs), which involve partial derivatives.
What they are-
An equation that contains an unknown function and one or more of its derivatives.
A solution is a function that satisfies the equation.
The goal is to find this function, which describes the relationship between a quantity and its rate of change.
Types-Ordinary Differential Equations (ODEs): Involve functions of a single independent variable and their ordinary derivatives.
Example: A simple model for population growth might be an ODE.
Partial Differential Equations (PDEs): Involve partial derivatives of a function of multiple independent variables.
Example: The equation describing how temperature changes over a solid body is a PDE.
Order: The order of a differential equation is determined by the highest derivative present in the equation. For example, an equation with a third derivative is a third-order ODE. Solution: A function that makes the differential equation true when substituted into it along with its derivatives. Initial conditions: To find a unique solution for an \(n^{th}\)-order differential equation, you typically need \(n\) initial conditions.
