Differential Equations
Study of equations involving derivatives and their applications.
Basic Concepts
Initial Value Problems
Starting from a Specific Point
Many problems give you an initial condition, like "when \( x = 0 \), \( y = 2 \)." This is called an initial value problem (IVP).
Why Are IVPs Important?
They allow you to find the specific solution that fits the starting condition, not just a general family of solutions.
Steps to Solve an IVP
- Solve the differential equation generally.
- Plug in the initial condition to find the constant.
- Write the particular solution with the constant's value.
Real-World Analogy
Imagine a rocket launching from the ground; knowing its position at launch helps predict its future path.
Examples
Solving \( \frac{dy}{dx} = 2x \) with \( y(0) = 1 \) gives \( y = x^2 + 1 \).
A cooling object at 100°C at \( t = 0 \), using Newton's law of cooling.
In a Nutshell
Initial value problems help us find the exact solution that fits a starting point.