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First-Order Differential Equations

Emily Davis

Emily Davis

7 min read

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Study Guide Overview

This study guide covers differential equations, focusing on first-order differential equations and their use in modeling real-world problems involving rates of change. It explores general and particular solutions, verifying solutions through differentiation, and includes initial conditions. Key concepts like derivatives and modeling are emphasized.

Modeling with Differential Equations

Table of Contents

  1. What is a Differential Equation?
  2. What is a First Order Differential Equation?
  3. Why are Differential Equations Useful for Modeling?
  4. General and Particular Solutions to Differential Equations
  5. Verifying Solutions to Differential Equations
  6. Practice Questions
  7. Glossary
  8. Summary and Key Takeaways

What is a Differential Equation?

A differential equation is an equation that involves derivatives of a function. It expresses the relationship between a function and its derivatives.

For example, dydx=12xy2\frac{dy}{dx}=12x{y}^{2} is a differential equation.

Another example is d2xdt2−5dxdt+7x=5sin⁡(t).\frac{d^2x}{dt^2} - 5\frac{dx}{dt} + 7x = 5\sin(t).

A differential equation includes both variables and the rates of change of those variables.

What is a First Order Differential Equation?

A first order differential equation is a differential equation that contains only first derivatives and no higher-order derivatives.

For example, dydx=12xy2\frac{dy}{dx}=12x{y}^{2} is a first order differential equation.

However, d2xdt2−5dxdt+7x=5sin⁡(t)\frac{d^2x}{dt^2} - 5\frac{dx}{dt} + 7x = 5\sin(t) is not a first order differential equation because it contains the second derivative d2xdt2\frac{d^2x}{dt^2}.

Why are Differential Equations Useful for Modeling?

Many real-world problems involve rates of change. Differential equations help us model these problems mathematically.

Some examples include: - The rate of change of a population of animals in a geographic area - The rate of change of the number of people infected by a particular disease - The rate of change of the amount of medication in a person's bloodst...
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