initial value

In mathematics, an initial value refers to a starting point or a set of values imposed on the variables within an equation that contains arbitrary constants. These values are crucial when solving differential equations, which are mathematical statements describing rates of change.12

When dealing with physical problems, differential equations often require auxiliary conditions to find a specific solution. These conditions can be either boundary values, which describe external influences on a system, or initial values, which provide a compressed history of the system's state at a specific time. For instance, in a differential equation describing motion, the initial position and velocity of an object would be its initial values. These starting points help determine the unique solution from a general set of possibilities.3