Φ(z2−xy,x/z)=0 (1)
We set u=z2−xy; v=zx.
Then the operator d on (1) leads to:
dΦ(u,v)=∂u∂Φ(u,v)du+∂v∂Φ(u,v)dv=0.
Thus
0=du=d(z2−xy) (3)
0=dv=d(zx) (4)
From (3) and (4) we have:
2zdz−ydx−xdy=0
z1dx−z2xdz=0
Thus getting rid of dx we obtain:
dx=zxdz
(2z−yzx)dz=xdy
dz=xz2z2−yxdy=xzx2dx
xzdz=2z2−yxdy=x2dx
Finally we obtain the desired PDE for z(x,y):
x2∂x∂z+(2z2−xy)∂y∂z=xz
Answer: x2∂x∂z+(2z2−xy)∂y∂z=xz.
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