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August 19th, 2017, 11:21 PM  #1 
Member Joined: Apr 2017 From: India Posts: 34 Thanks: 0  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. 
August 25th, 2017, 03:49 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
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. 
October 15th, 2017, 01:03 AM  #3 
Member Joined: Apr 2017 From: India Posts: 34 Thanks: 0 
The source I am referring to says it is differential equation with order 1 and degree 2. Well I think there is some sort of confusion. Could someone please elaborate that for me?. I am totally confused 
October 16th, 2017, 06:35 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 417 Thanks: 231 Math Focus: Dynamical systems, analytic function theory, numerics 
Indeed this is order 1 and degree 2. It is order 1 because it has 3 differential terms: $dx,dy,dz$ and each is a first order differential. It is degree 2 because the highest degree that any of these differentials appears is to the 2nd power (in this case all 3 are raised to the 2nd power). Country boy's post would be correct if $(dy)^2$ means $d(dy)$ but it doesn't. It means literally the square of the differential $(dy)^2 = (dy)\cdot (dy)$. 
October 17th, 2017, 01:42 AM  #5 
Member Joined: Apr 2017 From: India Posts: 34 Thanks: 0 
That clarified everything. Thank you so much.


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degree, differential, differential equation, equation, order 
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