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I'm working on Problem 8, Chapter 1 in the classic Ince text on ODEs.

The question has me somewhat confused.

It involves Euler's Theorem on Homogeneous functions. For reference, this theorem states that if you have a function f in two variables (x,y) and homogeneous in degree n, then you have:

[tex]x\frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = nf(x,y)[/tex]

The proof of this is straightforward, and I'm not going to review it here.

The problem asks the reader to prove the following theorem (I'm going to state the problem more or less verbatim, so I don't "read" anything into it that I'm not supposed). If f is a function homogeneous and of degree m in x1, x2, and homogeneous and of degree n in y1, y2, then:

[tex]\left(y_1 \frac{\partial}{\partial x_1} + y_2 \frac{\partial}{\partial x_2}\right)\left(x_1 \frac{\partial f}{\partial y_1} + x_2 \frac{\partial f}{\partial y_2}\right) - \left(x_1 \frac{\partial}{\partial y_1} + x_2 \frac{\partial}{\partial y_2} \right)\left(y_1 \frac{\partial f}{\partial x_1} + y_2 \frac{\partial f}{\partial x_2}\right) = (n-m)f[/tex]

In other words, the reader is asked to prove an extension to Euler's theorem.

I have some "attempts" at a solution, which I'll post in the next post, but to keep this post from growing too long and complicated, I'm going to end this one here.

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# Homework Help: Euler's Theorem, Homogenous Functions

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