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August 19th, 2017, 11:21 PM   #1
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Order and degree of a differential equation

I am unable to find the order and degree of the following differential equation:

$x^2(dx)^2 + 2xy dx dy + y^2(dy)^2 - z^2(dz)^2 = 0$

My approach:

Take the term $z^2(dz)^2$ to right hand side and then divide throughout by $(dz)^2$

we get $$\mathrm{\dfrac{x^2(dx)^2}{(dz)^2} + \dfrac{2xy\ dx \ dy}{(dz)^2} +\dfrac{y^2(dy)^2}{(dz)^2} = z^2}$$

By seeing this, we can deduce that order is 1 and degree 2. But when I am looking at the middle term it contains dx dy/(dz)^2 term which is a term of order 2 and hence degree 1. Well am I right here? Is there any other way of finding the degree and order of this total differential equation? Do suggest, if any.
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August 25th, 2017, 03:49 AM   #2
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Since "dx", "dy", and "dz" are differentials, you shouldn't interpret "$\displaystyle (dx)^2$", "$\displaystyle (dy)^2$", and "(dz)^2" as powers - those indicate second derivatives. This equation is of second order and first degree.

Last edited by skipjack; August 25th, 2017 at 06:49 AM.
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