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the solution looks like this ...

what kind of surface is that ... man i thought it would be a circle.

i exactly did the same thing and got the same result!!
x^2+y^2+z^2=0 !!

but this is wrong because it gives x=y=z=0

The lagrange method works ... but this is too ugly ... I have to eliminate z from 4th order equation

man i cant get that answer with charpit

@experimentX santosh where are u....lol

still here man ... fishing answer from another site.

what did you get for answer?
i got
z = a x + phi(a) y + c

had been doing
\[ z = k_1x+k_2y+k_3 \]

k2 should be some function of k1 ... i guess there it would reduce my trouble by half

man let me try again..

let me try it again too

I need to find the particular integral of the given DE passing through both of these curves.

no luck ... can't find \( \phi(k_1) \)

and substitute it there?

Sure

Looks like I messed up with Lagrange sol ... it didn't seem that difficult
\[z/x = \phi(y/x)\]