Talk:Homogeneous differential equation

Homogenous?
Is it homogenous or homogeneous because both are used in the article, it should be consistent. 130.95.128.51 (talk) 01:26, 23 May 2008 (UTC)

Two things
First, shouldn't we also note that it is also homogeneous if an equation of the form $$\frac{dy}{dx} = F(x,y).$$ can be rewritten as $$\frac{dy}{dx} = F(tx,ty)=t^nF(x,y).$$ for some n? Also, isn't an nth order differential equation with constant coefficients called homogeneous? I see no disambiguation of mention of that anywhere. Asmeurer ( talk   ♬  contribs ) 05:33, 14 March 2009 (UTC)

Intro completely rewritten
I've now rewritten the whole intro to the homogeneous differential equation article using an alternative formulation that makes the link to homogeneous functions clearer, rather than just taking the special case of f(x,y) = F(y/x), which only covers the case of homogeneous differential equations of degree 0. This is, in my opinion, both more elegant and more general. However, my maths education was a very long time ago: can people here please check my working? -- The Anome (talk) 03:52, 17 February 2010 (UTC)

And, alas, back again
I'm in complete agreement with Asmeurer, above, that the more general case is also called a homogeneous differential equation. The reference given in the article is a good one, but the overwhelming weight of all the references I can find online seems to support only the first case, so I've reluctantly wound the definition back to the previous case (although avoiding the F(y/x) presentation, with its singularity at x=0, which I think is confusing).

In particular, the second part of the original article clearly follows the more general definition, so it has, for the moment, to come out. I've pasted it below so that it can be restored as and when we have more references for the more general definition.

A well known homogenous equation in x and y, subsequently showing one of Euler's identities, is as follows.


 * $$\ f(x,y) = x^n F\left(\frac{y}{x}\right).$$

This is clearly a homogeneous equation of degree n, since, by inspection:


 * $$\ {(tx)}^n F\left(\frac{ty}{tx}\right) = t^nx^n F\left(\frac{y}{x}\right)$$

Differentiating$$\ f_x (x,y) $$ we obtain the following,


 * $$\frac{\partial f(x,y)}{\partial x} = nx^{n-1}F\left(\frac{y}{x}\right)+ x^nF^'\left(\frac{y}{x}\right)\cdot\left(-\frac{y}{x^2}\right)$$.

Where $$\ F^'$$ denotes the first derivative of F with respect to the homogeneous argument.

Also,


 * $$\frac{\partial f(x,y)}{\partial y} = x^nF^'\left(\frac{y}{x}\right).\left(\frac{1}{x}\right).$$

Now taking each derivative and multiplying by its corresponding variable we arrive at the following equation.


 * $$ x\frac{\partial f(x,y)}{\partial x} + y\frac{\partial f(x,y)}{\partial y} = x\left[nx^{n-1}F\left(\frac{y}{x}\right)+ x^n.\left(-\frac{y}{x^2}\right)F^'\left(\frac{y}{x}\right)\right] + y\left[ x^n.\left(\frac{1}{x}\right)F^'\left(\frac{y}{x}\right)\right]$$


 * $$ x\frac{\partial f(x,y)}{\partial x} + y\frac{\partial f(x,y)}{\partial y} = x\left[nx^{n-1}F\left(\frac{y}{x}\right)-x^{n-2}yF^'\left(\frac{y}{x}\right)\right] + y\left[ x^{n-1}F^'\left(\frac{y}{x}\right)\right]$$


 * $$= nx^nF\left(\frac{y}{x}\right)$$


 * $$\ = nf(x,y).$$

Which in turn is one of Euler's identities,


 * $$x\frac{\partial f(x,y)}{\partial x} + y\frac{\partial f(x,y)}{\partial y} = nf(x,y).$$

This identity is generalized by Euler's theorem on homogeneous functions. -- The Anome (talk) 14:18, 20 February 2010 (UTC)

Can homogeneous and inhomogeneous also describe systems of linear algebraic equations.
Hi- I am working on this Wikiversity lesson that involves linear equations of the form


 * $$A_{ij}x_j=b_i$$ (summation over repeated indexes is implied)

I always thought this was called an "inhomogeneous" set of equations. I also saw the same terminology used in this MIT pdf file:

http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-b-matrices-and-systems-of-equations/session-14-solutions-to-square-systems/MIT18_02SC_MNotes_m3.pdf

---guyvan52 (talk) 21:51, 27 November 2014 (UTC)


 * After posting I noticed your title "inhomogeneous differential equations", which explains why you omitted mention of the algebraic equations. I had both pages open on my browser and posted on the wrong one. I will take this question to the talk page for System of linear equations-- Mea culpa--guyvan52 (talk) 22:05, 27 November 2014 (UTC)