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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

Step 1: Rearrange the Equation

Rearrange the differential equation into the form:

(1h(y))dydx=g(x)\left(\frac{1}{h(y)}\right)\frac{dy}{dx} = g(x)

For the equation dydx=−xy3\frac{dy}{dx} = -xy^3:

−1y3dydx=x-\frac{1}{y^3} \frac{dy}{dx} = x

Here, g(x)=xg(x) = x and h(y)=−y3h(y) = -y^3.

Step 2: Integrate Both Sides

Integrate both sides with respect to xx:

∫1h(y)dy=∫g(x)dx\int \frac{1}{h(y)} dy = \int g(x) dx

For the equation −1y3dydx=x-\frac{1}{y^3} \frac{dy}{dx} = x:

∫−1y3dy=∫xdx\int \frac{-1}{y^3} dy = \int x dx

Think of this step as 'multiplying the dxdx across and integrating both sides'. While this is not mathematically precise, it helps in understanding the process.

Step 3: Solve the Integrals

Evaluate the integrals on both sides of the equation. Don't forget to include a constant of integration:

∫−1y3dy=∫xdx⇒−12y2=x22+C\int \frac{-1}{y^3} dy = \int x dx \Rightarrow -\frac{1}{2y^2} = \frac{x^2}{2} + C

You only need one constant of integration even though there are two integrals.

Step 4: Rearrange the Solution

Rearrange the solution to isolate yy:

−12y2=x22+C⇒1y2=−x2−2C⇒y2=1−x2−2C-\frac{1}{2y^2} = \frac{x^2}{2} + C \Rightarrow \frac{1}{y^2} = -x^2 - 2C \Rightarrow y^2 = \frac{1}{-x^2 - 2C}

Note that 2C2C is just another arbitrary constant, so it can be written as CC:

y2=1−x2+Cy^2 = \frac{1}{-x^2 + C}

For the equation dydx=ex+4x3y2\frac{dy}{dx} = \frac{e^x + 4x}{3y^2}:

3y2dydx=ex+4x3y^2 \frac{dy}{dx} = e^x + 4x

Integrate both sides:

∫3y2dy=∫(ex+4x)dx\int 3y^2 dy = \int (e^x + 4x) dx

Perform the integrations:

y3=ex+2x2+Cy^3 = e^x + 2x^2 + C

Rearrange to obtain yy:

y=ex+2x2+C3y = \sqrt[3]{e^x + 2x^2 + C}

Worked Example

Problem: Use separation of variables to solve the differential equation:

dydx=ex+4x3y2\frac{dy}{dx} = \frac{e^x + 4x}{3y^2}

Solution:

  1. Separate the variables:

3y2dydx=ex+4x3y^2 \frac{dy}{dx} = e^x + 4x

  1. Integrate both sides with respect to xx:

∫3y2dy=∫(ex+4x)dx\int 3y^2 dy = \int (e^x + 4x) dx

  1. Compute the integrals:

y3=ex+2x2+Cy^3 = e^x + 2x^2 + C

  1. Rearrange to express yy:

y=ex+2x2+C3y = \sqrt[3]{e^x + 2x^2 + C}

Practice Questions

Practice Question

Question 1: Solve the differential equation dydx=y⋅cos⁡(x)\frac{dy}{dx} = y \cdot \cos(x).

Practice Question

Question 2: Solve the differential equation dydx=2xy+1\frac{dy}{dx} = \frac{2x}{y + 1}.

Practice Question

Question 3: Solve the differential equation dydx=(x2+1)y\frac{dy}{dx} = (x^2 + 1)y.

Glossary

  • Differential Equation: An equation involving derivatives of a function.
  • Separable Differential Equation: A differential equation that can be written in the form dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y).
  • Integration Constant: An arbitrary constant added to the function when performing indefinite integration.

Summary and Key Takeaways

Summary

  • Separation of variables is a technique for solving first-order differential equations by separating xx and yy terms.
  • The process involves rearranging the equation, integrating both sides, and solving for the function.
  • The method provides a general solution which can further be specified to a particular solution if initial conditions are provided.

Key Takeaways

  • Recognize the form dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y) to apply separation of variables.
  • Always include a constant of integration when solving integrals.
  • Rearrange the final solution to the desired form if required.
Exam Tip

Practice recognizing separable differential equations and applying these steps to gain proficiency.

Exam Strategy

  • Carefully check if a given differential equation is separable.
  • Make sure to include the constant of integration.
  • If the exam requires a particular form of the solution, ensure to rearrange accordingly.

Understanding the separation of variables technique is crucial for solving many first-order differential equations in exams.

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