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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?

Please, do tell me.

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"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. –  Guess who it is. Dec 21 '11 at 5:21
    
Thanks. ......... –  VVV Dec 21 '11 at 5:23

4 Answers 4

up vote 5 down vote accepted

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.

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I like this answer, thank you a lot. –  VVV Dec 21 '11 at 5:23
    
This was a very good explanation!!! Cleared up a bit for sure! –  user41920 Sep 20 '12 at 14:09

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.

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could you tell how the term 'homogeneous' came in mathematics? –  justin Nov 13 '14 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". –  Hasan Saad May 12 at 10:13
    
@HasanSaad:Could you give an example.Is it only for linear equations? –  justin May 12 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 $c$. –  Hasan Saad May 12 at 10:20
    
@HasanSaad:Could you give an simple example with equations without log or derivatives or any calculus or exponential terms. –  justin May 12 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.

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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

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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. –  G. Sassatelli May 12 at 10:16

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