Differential Equations

Study of equations involving derivatives and their applications.

What are Differential Equations?First-Order Differential EquationsInitial Value Problems
Second-Order Differential EquationsSystems of Differential EquationsPartial Differential Equations (PDEs)
Modeling Population GrowthPhysics: Motion and ForcesMedicine: Drug Dosage and Decay
Break Down the ProblemCheck Your Solution
What are Differentia...First-Order Differen...Initial Value Proble...Second-Order Differe...Systems of Different...Partial Differential...Modeling Population ...Physics: Motion and ...Medicine: Drug Dosag...Break Down the Probl...Check Your Solution
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

  1. Solve the differential equation generally.
  2. Plug in the initial condition to find the constant.
  3. 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.

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Second-Order Differential Equations