Question

2 Solve x(y²-z²)p-y(z²+x²)q=z(x²+y²)​

Answer 1

Answer:

Solve the PDE

x(y2+z)p−y(x2+z)q=(x2−y2)z

where, p=∂z∂x and q=∂z∂y

My attempt:

I start with Lagrange's auxiliary equation

dxx(y2+z)=dy−y(x2+z)=dzz(x2−y2)

Relation 1:

xdx+ydy−dzx2y2+x2z−y2x2−y2z−x2z+y2z=xdx+ydy−dz0

Integrating xdx+ydy−dz=0

x22+y22−z=c⟹x2+y2−2z=c1