# How to tell if a differential equation is homogeneous, or inhomogeneous?

Sometimes it arrives to me that I try to solve a linear differential equation for a long time and in the end it turn out that it is not homogeneous in the first place.

Is there a way to see directly that a differential equation is not homogeneous?

• "Homogeneous" means that the only entities present are the unknown function and its derivatives (possibly with some coefficients). Thus $y''=xy$ is homogeneous; $y''=xy+x+1$ is not, since $x+1$ doesn't "involve" $y$ or its derivatives. Dec 21, 2011 at 5:21
• homogeneous means you can prove the space of solutions is a vector space with your eyes closed in $2$ seconds. Nov 27, 2015 at 19:30

For a linear differential equation $$a_n(x)\frac{d^ny}{dx^n}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_1(x)\frac{dy}{dx}+a_0(x)y=g(x),$$ we say that it is homogenous if and only if $g(x)\equiv 0$. You can write down many examples of linear differential equations to check if they are homogenous or not. For example, $y''\sin x+y\cos x=y'$ is homogenous, but $y''\sin x+y\tan x+x=0$ is not and so on. As long as you can write the linear differential equation in the above form, you can tell what $g(x)$ is, and you will be able to tell whether it is homogenous or not.

• I like this answer, thank you a lot.
– VVV
Dec 21, 2011 at 5:23
• This was a very good explanation!!! Cleared up a bit for sure!
– user41920
Sep 20, 2012 at 14:09
• Finally clicked! Thank you. Nov 4, 2020 at 3:42

The simplest test of homogeneity, and definition at the same time, not only for differential equations, is the following:

An equation is homogeneous if whenever $\varphi$ is a solution and $\lambda$ scalar, then $\lambda\varphi$ is a solution as well.

• could you tell how the term 'homogeneous' came in mathematics? Nov 13, 2014 at 9:12
• @justin I am not sure about terminology, but it seems because all scalar multiples of a solution are solutions and thus they "mix" as "one" not "separately". May 12, 2015 at 10:13
• @HasanSaad:Could you give an example.Is it only for linear equations? May 12, 2015 at 10:17
• I am not sure it holds for non-linear ones, but for linear ones, you can easily just plug $\lambda\phi$ in the equation and it "gets out" of the derivatives, so it'll work. As for an example, $y'+y=0$, then $y=ce^{-x}$ holds for all scalars $c$. May 12, 2015 at 10:20
• @HasanSaad:Could you give an simple example with equations without log or derivatives or any calculus or exponential terms. May 12, 2015 at 12:05

A homogeneous differential equation have same power of $X$ and $Y$ example :$- x+y dy/dx= 2y$

$X+y$ have power $1$ and $2y$ have power $1$ so it is an homogeneous equation.

The best and the simplest test for checking the homogeneity of a differential equation is as follows :--> The formulae is $$\frac{dy}{dx}=F(x,y)$$ such that $$F(x,y)=F(tx,ty)$$ for "t" being any arbitrary constant, then $$\frac{dy}{dx}$$ is homogeneous.

Take for example we have to solve $$\frac{dy}{dx}=\frac{y+ (x^2+y^2)^\frac12}{x}$$

1. Put $$x=tx$$ and $$y=ty$$ where t is any arbitrary constant.

2. Now from the numerator and denominator take the constant as common with maximum power possible from both numerator and denominator each.

3. If the constant gets cancelled throughout and we obtain the same equation again then that particular differential equation is homogeneous and the the power of constant which remains after cutting it to lowest degree is the degree of homogeneity of that equation. Hope That helped!

• What's the theory that supports this process you're suggesting here @aditya2222? Jun 1, 2018 at 16:38
• What's happens if $\alpha<0$. I though a possible problem with the definition because $\sqrt{\alpha^2}=|\alpha|=-\alpha\not=\alpha$. Jul 12, 2019 at 14:25

Any function like y and its derivatives are found in the DE then this equation is homgenous

ex. y"+5y´+6y=0 is a homgenous DE equation

But y"+xy+x´=0 is a non homogenous equation becouse of the X funtion is not a function in Y or in its derivatives

• Well, it seems to me that you are a bit confused on either the definition of "homogeneous" or the explanation of your thoughts. Please, take your time to better elaborate your post. Also, having a look at other people's answers might help.
– user228113
May 12, 2015 at 10:16

I think a differential equation is homogeneous if every term contains y or derivatives of y in the equation

• Please, do not answer to four-year-old questions unless you are adding something substantial to the discussion.
– user228113
Nov 27, 2015 at 20:05

if you are given an ODE say $f(x,y)=x^2-3xy+5y^2$ and they ask you to show if it is homogeneous or not here is how to do it

If a function $f$ has the property that $f(tx,ty)=t^nf(x,y)$ then the function is homogeneous of degree n...to prove if the above DE is homogeneous here is how to do it \begin{align*} f(x,y)&=x^2-3xy+5y^2\\ f(tx,ty)&=(tx)^2-3(tx)(ty)+5(ty)^2\\ &=t^2x^2-3t^2xy+5t^2y^2\text{ if we factor $t^2$ we get}\\ &=t^2[x^2-3xy+5y^2]\\ &=t^2f(x,y) \end{align*} hence the function is homogeneous of degree $n$

• Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Jul 12, 2018 at 9:17
• The question is 6½ years old, and has an accepted answer, so not many people will see this answer. It also only seems to address a very specific DE, where the question is general, so it seems like a waste of time that would be better spend on something else e.g. learning MathJax, @JoséCarlosSantos has posted a link. Jul 12, 2018 at 9:39

The equations in the form $$f(xy)$$ can be said to be homogeneous also if they can be put in the form $$dy/dx =f(y/x)$$ or in other cases $$f(x,y )=x^n g(y/x)$$