Differential Equations - Homogeneous

When a first order differential equation is not separable, nor linear (integrating factor), it may still be possible to solve it analytically using a substitution. This will work when the equation is homogeneous. A homogenous differential equation is a differential equation that can be written as a function of \(\large \frac{y}{x}\). We use a variable substitution \(\large (v=\frac{y}{x})\) and this turns our equation into a variables separable differential equation


Key Concepts

On this page, you should learn about

  • recognising homogeneous differential equations in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}=f(\frac{y}{x})\)
  • solving homogeneous differential equations using the substitution \(y=vx\)

Essentials

The process for solving homogenous differential equations is always the same. However, it is quite a long one! The following video will help you understand what you need to know

Example question

In the following video we look at how we can solve homogenous differential equations. The process is quite long, there is a lot of algebraic manipulation, so the video is quite long. Grab a pen and work along with the video at the same time!


Use the substitution \(\large y=vx\) to solve the differential equation

\(\large 3xy^2\frac{\text{d}y}{\text{d}x}=x^3+y^3\)

Notes from the video

Summary

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Test Yourself

The following quiz will give you some practice in recognising the type of differential equation to be solve: variables separable, integrating factor or homogeneous


START QUIZ!

Exam-style Questions

Question 1

Consider the first order differential equation

\( \large xy\frac{\text{d}y}{\text{d}x}+4x^2+y^2=0 \quad,\quad y\geq0\)

a) Use the substitution y = vx to show that \( \large \frac{\text{d}v}{\text{d}x}=-\frac{4+2v^2}{vx}\)

b) Show that the solution to the differential equation for which y(1) = 0 is \(\large y=\frac{\sqrt{2-2x^4}}{x}\)

Hint

Full Solution

 

Question 2

Consider the homogeneous differential equation \(\large \frac{\text{d}y}{\text{d}x}=\frac{x^2+y^2}{2xy}\)

a) Using the substitution y = vx , show that \(\large \frac{\text{d}v}{\text{d}x}=\frac{1-v^2}{2vx}\)

b) Solve the differential equation and show that y² = x² - cx

Hint

Full Solution

 

Question 3

Consider the differential equation \(\large xy\frac{\text{d}y}{\text{d}x}=x^2\text{cosec}\frac{y}{x}+y^2\)

a) Show that this equation can be written in the form \(\large \frac{\text{d}y}{\text{d}x}=f(\frac{y}{x})\)

b) Using the substitution y = vx , show that the particular solution to the equation, given that y(1)= 0 is

Hint

Full Solution

 

MY PROGRESS

How much of Differential Equations - Homogeneous have you understood?