Calculus – Initial Value Problems

Book Calculus by Thomas/Finney, 9th Edition, Page 282

Understanding Initial Value Problems in Calculus

When studying differential equations in calculus, we often encounter something called an Initial Value Problem (IVP). While differential equations describe entire families of functions, an initial value problem helps us find one specific solution that fits a given condition.

What Is an Initial Value Problem?

An Initial Value Problem consists of two parts:

1. A differential equation
2. An initial condition, which specifies the value of the function at a particular point

In general form:
$$\frac{dy}{dx} = f(x, y), \quad y(x_0) = y_0$$​

The differential equation describes how a function changes, while the initial condition tells us where the solution starts.

Example.