# Calculus

### Chapters

### Homogeneous Differential Equations

# Homogeneous Differential Equations

## Introduction

`Differential Equations`

are equations involving a function and one or more of its derivatives.

For example, the differential equation below involves the function \(y\) and its first derivative \(\dfrac{dy}{dx}\).

Let's consider an important real-world problem that probably won't make it into your calculus text book:

A plague of feral caterpillars has started to attack the cabbages in Gus the snail's garden. Gus observes that the cabbage leaves are being eaten at the rate

Gus has written down a

`homogeneous`

differential equation. We'll find out what this means in a minute.
To work out how many cabbage leaves will be eaten on day \(2\), he needs to solve his differential equation and plug in \(t=2\). But how can he solve
the equation?
## What are Homogeneous Differential Equations?

A first order differential equation is `homogeneous`

if it can be written in the form:

For example, the differential equation

## Solving Homogeneous Differential Equations

We need to transform these equations into separable differential equations. We begin by making the substitution \(y = vx\). Differentiating gives

### Example

First, check that it is homogeneous. Let \(k\) be a real number. Then

Next, do the substitution \(y = vx\) and \(\dfrac{dy}{dx} = v + x \; \dfrac{dv}{dx}\):

Now, apply separation of variables:

**Step 1:** Separate the variables by moving all the terms in \(v\), including \(dv\),
to one side of the equation and all the terms in \(x\), including \(dx\), to the other.

**Step 2:** Integrate both sides of the equation.

**Step 3:** There's no need to simplify this equation.

Now substitute \(y = vx\), or \(v = \dfrac{y}{x}\) back into the equation:

Let's try another.

### Example

First, check that it is homogeneous. Let \(k\) be a real number. Then

Next, do the substitution \(y = vx\) and \(\dfrac{dy}{dx} = v + x \; \dfrac{dv}{dx}\) to convert it into a separable equation:

Now, apply separation of variables:

**Step 1:** Separate the variables by moving all the terms in \(v\), including \(dv\),
to one side of the equation and all the terms in \(x\), including \(dx\), to the other.

**Step 2:** Integrate both sides of the equation.

**Step 3:** Simplify this equation. First, write \(C = \ln(k)\), and then
take exponentials of both sides to get rid of the logs:

I think it's time to deal with the caterpillars now.

### Solving Gus' Feral Caterpillar Problem

If you recall, Gus' garden has been infested with caterpillars, and they are eating his cabbages. He's modelled the situation using the differential equation:

First, we need to check that Gus' equation is homogeneous. Let \(k\) be a real number. Then \( \dfrac{k\text{cabbage}}{kt} = \dfrac{\text{cabbage}}{t}, \) so it certainly is!

Next do the substitution \(\text{cabbage} = vt\), so \( \dfrac{d \text{cabbage}}{dt} = v + t \; \dfrac{dv}{dt}\):

Finally, plug in the initial condition to find the value of \(C\) We plug in \(t = 1\) as we know that \(6\) leaves were eaten on day \(1\)

On day \(2\) after the infestation, the caterpillars will eat \(\text{cabbage}(2) = 6(2) = 12 \text{ leaves}.\) Poor Gus!

### Description

Calculus is the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are *differential calculus* and *integral calculus*.

### Environment

It is considered a good practice to take notes and revise what you learnt and practice it.

### Audience

Grade 9+ Students

### Learning Objectives

Familiarize yourself with Calculus topics such as Limits, Functions, Differentiability etc

Author: Subject Coach

Added on: 23rd Nov 2017

`You must be logged in as Student to ask a Question.`

None just yet!