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I'm reading the book Ordinary Differential Equations and Dynamical Systems by Gerald Teschl. Or rather, I am reading the online edition:

On page 15 it says:

More generally, consider the differential equation $$ \dot{x} = f\left(\frac{ax + bt + c}{\alpha x + \beta t + \gamma}\right). $$ Two cases can occur. If $a \beta - \alpha b = 0$, our differential equation is of the form $$\dot{x} = \widetilde{f}(ax +bt) $$

I understand the rest of the page. But this step makes no sense to me. Would someone care to explain to me what the author might mean?

Thanks in advance

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up vote 3 down vote accepted

$\beta = \frac{\alpha b}{a}$

plug in to get

$\dot{x} = f\left(\frac{a x + bt + c}{\alpha x + \alpha b/a t + \gamma}\right) = f\left(\frac{a x + bt + c}{\alpha/a( a x + b t + a\gamma/\alpha)}\right)$

which is a function of $ax + bt$.

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Thank you, that seems correct :) – DoubleTrouble Feb 2 '13 at 20:33

Since $a\beta - \alpha b$ is a determinant, and in the special case $c=\gamma=0$ you can always write the argument of $f$ in a $ax+bt$ form (because the numerator is a multiple of the denominator). In general, if the numerator is a multiple of the denominator, you could do a change of coordinates and resort the argument to a $ax+bt$ form, but seems to me that the condition $a\beta -\alpha b=0$ is necessary but not sufficient.

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