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First-Order Differential Equations

Emily Davis

Emily Davis

5 min read

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Study Guide Overview

This guide covers the separation of variables method for solving first-order differential equations. It outlines the steps involved: rearranging the equation into the form dy/dx = g(x)h(y), integrating both sides, and solving for y. The guide includes a worked example, practice questions, and a glossary of key terms like separable differential equations and integration constant. It also provides tips and strategies for applying this technique effectively in exams.

Separation of Variables

Table of Contents

  1. Introduction to Separation of Variables
  2. Steps to Solve Differential Equations Using Separation of Variables
  3. Worked Example
  4. Practice Questions
  5. Glossary
  6. Summary and Key Takeaways

Introduction to Separation of Variables

Separation of Variables is a method used to solve certain types of first-order differential equations. These equations typically take the form:

dydx=g(x)h(y)\frac{dy}{dx} = g(x) h(y)

where the derivative dydx\frac{dy}{dx} is equal to a function of xx, denoted as g(x)g(x), multiplied by a function of yy, denoted as h(y)h(y).

Exam Tip

Look out for equations that can be written in this form; this is key to applying the method effectively.

Key Concept

Note that the function of xx, g(x)g(x), can sometimes be a constant. For example, in the equation dydx=6y\frac{dy}{dx} = 6y, g(x)=6g(x) = 6 and h(y)=yh(y) = y.

Common Mistake

Ensure that the equation is correctly identified as separable. Misidentifying the functions g(x)g(x) and h(y)h(y) can lead to incorrect solutions.

Steps to Solve Differential Equations Using Separation of Variables...

Previous Topic - Slope FieldsNext Topic - Finding Particular Solutions

Question 1 of 9

Which of the following differential equations is separable? 🤔

dydx=x+y\frac{dy}{dx} = x + y

dydx=xy\frac{dy}{dx} = xy

dydx=x2+y2\frac{dy}{dx} = x^2 + y^2

dydx=x+2y\frac{dy}{dx} = x + 2y