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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. –  J. M. Dec 21 '11 at 5:21
    
Thanks. ......... –  VVV Dec 21 '11 at 5:23

2 Answers 2

up vote 2 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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