Is the Cauchy Equation Still Homogeneous if X is a Non-zero Function of x?

[iVenky]

The Cauchy homogeneous linear differential equation is given by

x^{n}\frac{d^{n}y}{dx^{n}} +k_{1} x^{n-1}\frac{d^{n-1}y}{dx^{n-1}} +...+k_{n}y=Xwhere X is a function of x and k_{1},k_{2}...,k_{n} are constants.

I thought for this equation to be homogeneous the right side should be 0. (i.e.) X=0.
But if X is a function of x then how can this be homogeneous?

Thanks a lot :)

 

[HallsofIvy]

It isn't. Why do you call it "homogeneous"?

(Googling "Cauchy's homogeneous equation", I found a "youtube" tape calling this equation "homogeneous"- its just wrong! I suspect they started talking about a homogeneous equation and did not change the title when they generalized).

 

[JJacquelin]

Let y(x) = Y(x)+(X/k_n) and the rigth side will be 0.

 

[iVenky]

Let y(x) = Y(x)+(X/k_n) and the rigth side will be 0.

If I do things like that, I can make any equation homogeneous.

Don't forget that X is a function of 'x'.

 

[HallsofIvy]

You are assuming that X is a constant, aren't you?

 

[JJacquelin]

Sorry, I was assuming that X was constant.
So, my answer is out of subject.

 

[iVenky]

I have chosen that X to be a function of 'x' and it is not a constant

So I think it is not homogeneous.

For the record even if X is a constant it is still not homogeneous, isn't it?

thanks a lot :)

 

[HallsofIvy]

I have chosen that X to be a function of 'x' and it is not a constant

So I think it is not homogeneous.

For the record even if X is a constant it is still not homogeneous, isn't it?

thanks a lot :)

Yes, that is correct.