The condition of integrability:
P(∂z∂Q−∂y∂R)+Q(∂x∂R−∂z∂P)+R(∂y∂P−∂x∂Q)=0
(yx2z2−y2z3)(x2y2−2zx3−2yxz2+3y2x2)+(zx2y2−z2x3)(y2z2−2xy3−
−2zyx2+3z2y2)+(xy2z2−x2y3)(x2z2−2yz3−2xzy2+3x2z2)=
=y3x4z2−2yx5z3−2y2x3z4+3y3x4z2−x2y4z3+2x3y2z4+2xy3z5−3x2y4z3+
+x2y4z3−2x3y5z−2x4y3z2+3x2y4z3−x3y2z4+2x4y3z2+2x5yz3−3x3y2z4+
+x3y2z4−2xy3z5−2x2y4z3+3x3y2z4−x4y3z2+2x2y4z3+2x3y5z−3x4y3z2=
=0
The equation is integrable.
The equation is homogeneous.
D=Px+Qy+Rz=xyz2(x2−yz)+yzx2(y2−zx)+zxy2(z2−xy)=
=x3yz2−xy2z3+x2y3z−x3yz2+xy2z3−x2y3z=0
Let
x=uz,y=vz
dx=udz+zdu,dy=vdz+zdv
Then:
vz3(u2z2−vz2)(udz+zdu)+z3u2(v2z2−z2u)(vdz+zdv)+
+uv2z3(z2−uvz2)dz=0
(vu3−v2u)dz+(vu2z−v2z)du+(v3u2−vu3)dz+(zv2u2−zu3)dv+
+(uv2−u2v3)dz=0
vz(u2−v)du+u2z(v2−u)dv=0
v(u2−v)du+u2(v2−u)dv=0
dudv=u2(u−v2)v(u2−v)
v=2u(c+1)2−4u3+cu+1
zy=2xz((c+1)2−4(x/z)3+(cx/z)+1)
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