A differential equation is an equation that involves a function and its derivatives. It represents the relationship between the function and its rate of change.
Solving a differential equation means finding a function or set of functions that satisfy the equation. This involves finding the general solution and then applying initial conditions to find the particular solution that satisfies the given initial value.
An initial value is a set of conditions that specify the value of the function and its derivatives at a particular point. These values are used to find the particular solution to a differential equation.
There are several methods for solving differential equations, including separation of variables, integrating factors, and using power series or Laplace transforms. The method used depends on the type of differential equation and its complexity.
Differential equations are important in science because they can model many physical, chemical, and biological processes. They allow scientists to make predictions and understand the behavior of systems in various fields such as physics, engineering, and biology.