pde question quasi lineer equation

**sah_mat** · February 19th 2009, 12:08 AM

QUESTİON:
x(y^2+z)z_x - y(x^2+z)z_y =(x^2-y^2).z
thanks for your helps dear friends.

**Danny** · February 19th 2009, 09:47 AM

Originally Posted by sah_mat

QUESTİON:
x(y^2+z)z_x - y(x^2+z)z_y =(x^2-y^2).z
thanks for your helps dear friends.

The characteristic equations are

$\frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}<br />$

from which we can deduce

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

and

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

giving the solution as

$x^2+y^2-2z = f(xyz)$

**sah_mat** · February 19th 2009, 11:12 AM

thanks for your solution,but one point ı am with trouble i did not understand your deduction which you did at here $\frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}$

to get this $\frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} =$
if you explain a little more ı will be gladfull to you.

**Danny** · February 19th 2009, 11:39 AM

Originally Posted by sah_mat

thanks for your solution,but one point ı am with trouble i did not understand your deduction which you did at here $\frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}$

to get this $\frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} =$
if you explain a little more ı will be gladfull to you.

Sure, let

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

or

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

so

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

and if you add

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

or what I wrote earlier.

**sah_mat** · February 19th 2009, 10:59 PM

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.

**Danny** · February 20th 2009, 05:04 AM

Originally Posted by sah_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.

Before I can help, we need to find common ground. Please solve the following so I ca nsee what your teacher taught you

$x z_x + y z_y = 0$ ,

$x z_x +(x+y) z_y = z$ ,

$z_x + z z_y = x$ .

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