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I've been working through some problems in my college differential equations textbook, and I've come to one which asks for a proof that if $f(x, y)$ is a homogenous function of degree $n$, then:

$$ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f $$

I started with the definition of a homogeneous function:

$$ f(tx, ty) = t^n f(x, y) $$

To get a factor of $n$ somewhere, the obvious thing to try is to differentiate with respect to $t$:

$$ \begin{align} \frac{\partial f}{\partial x} \frac{\partial}{\partial t}(tx) + \frac{\partial f}{\partial y} \frac{\partial}{\partial t}(ty) & = n t^{n-1} f(x, y)\\ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} & = n t^{n-1} f(x, y) \end{align} $$

That's almost in the right form except for that pesky factor of $t^{n-1}$. Where did I go wrong?

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  • $\begingroup$ Note that in the left hand side you evaluate $\frac{\partial f}{\partial x}(tx,ty)$. For $t=1$ this is exactly what you want. $\endgroup$
    – Or Kedar
    Commented Jun 15, 2018 at 15:50

1 Answer 1

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Hint: Fix $(x,y)$ and let $g(t) = f(tx,ty).$ There are two ways to compute $g'(1).$

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