QUESTİON:

x(y^2+z)z_x - y(x^2+z)z_y =(x^2-y^2).z

thanks for your helps dear friends.

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- Feb 19th 2009, 12:08 AMsah_matpde question quasi lineer equation
QUESTİON:

x(y^2+z)z_x - y(x^2+z)z_y =(x^2-y^2).z

thanks for your helps dear friends. - Feb 19th 2009, 09:47 AMJester
The characteristic equations are

$\displaystyle \frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}

$

from which we can deduce

$\displaystyle \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} = \;\;\; \Rightarrow\;\;\; xyz = c_1$

and

$\displaystyle x\, dx + y\, dy - dz = 0 \;\;\; \Rightarrow\;\;\; x^2 + y^2 -2 z = c_2$

giving the solution as

$\displaystyle x^2+y^2-2z = f(xyz)$ - Feb 19th 2009, 11:12 AMsah_mat
thanks for your solution,but one point ı am with trouble i did not understand your deduction which you did at here $\displaystyle \frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}$

to get this $\displaystyle \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} = $

if you explain a little more ı will be gladfull to you. - Feb 19th 2009, 11:39 AMJester
Sure, let

$\displaystyle \frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}$

or

$\displaystyle \frac{dx}{dt} =x(y^2+z),\;\; \frac{dy}{dt} = -y(x^2+z),\;\; \frac{dz}{dt} = (x^2- y^2)z $

so

$\displaystyle \frac{1}{x}\,\frac{dx}{dt} =y^2+z,\;\; \frac{1}{y}\, \frac{dy}{dt} = -(x^2+z),\;\; \frac{1}{z}\, \frac{dz}{dt} = x^2- y^2 $

and if you add

$\displaystyle \frac{1}{x}\,\frac{dx}{dt} + \frac{1}{y}\,\frac{dy}{dt} + \frac{1}{x}\,\frac{dx}{dt} = 0$

or what I wrote earlier. - Feb 19th 2009, 10:59 PMsah_mat
i discovered that u are solving wşth a different way from my teacher taught me,and i am at the start of pde hence i have some misunderstandable points left about your solution if you let me to ask you i will be happy,

how did you found c1 and c2 please explain me all steps ,than i will be capable of solving other questions with your way which is more understandable and easy to understand.thanks a lot for oyur helps danny. - Feb 20th 2009, 05:04 AMJester