I have to solve:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

x(y^2 - z^2) \frac{\partial z}{\partial x} + y(z^2 - x^2) \frac{\partial z}{\partial y} + z(x^2 - y^2)

[/tex]

So, I write out the characteristic system of ODEs:

[tex]

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

[/tex]

Now, the variables aren't seperated so I can't integrate two pairs seperately, so what I did was use the componendo et dividendo rule for fractions to write :

[tex]

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

\frac{yzdx+xzdy+xydz}{xyz(y^2 - z^2)+xyz(z^2 - x^2)+xyz(x^2 - y^2)} =

\frac{yzdx+xzdy+xydz}{0}

[/tex]

Thus

[tex]

yzdx+xzdy+xydz=0

[/tex]

which is in integrable Pfaffian Equation, and its integration yields:

[tex]

xyz=C_1

[/tex]

ie. I have one of the constants of integration of the characteristic system of ODEs.

But now I'm stuck because I don't know how to obtain the other :S

I thought to use one pair of the ODE system, eg,

[tex]

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

[/tex]

and substitute

[tex]

z=C_1/(xy)

[/tex]

from the first integration, so that I'd be left with an ode:

[tex]

\frac{dx}{x(y^2 - C_1^2/(xy)^2)} = \frac{dy}{y(C_1^2/(xy)^2) - x^2)}

[/tex]

from which in theory we could obtain a second integration constant, but this is too hard to solve (I think), so there must be an easier way to get the second integration constant, ie.

[tex]

g(x,y,z)=C_2

[/tex]

And the general solution of the PDE would be

[tex]

F[xyz,g(x,y,z)]=0,

[/tex]

with F an arbitrary function.

Anyone have any ideas please?:P

Thank you :)

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# Help with 1st order quasilinear PDE

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