Thread: pde question quasi lineer equation

1. pde 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.

2. 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

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

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

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

3. 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.

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

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.
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

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

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

or what I wrote earlier.

5. 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.

6. 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

$\displaystyle x z_x + y z_y = 0$,

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

$\displaystyle z_x + z z_y = x$.