91Ó°ÊÓ

Find the complete integral of the differential equation $$ x p(1+q)=(y+z) q $$ corresponding to that integral of Charpit's equations which involves only \(q\) and \(x\), and deduce that $$ (z+h x+k)^{2}=4 h x(k-y) $$ is also a complete integral.

If any integral surface of a partial differential equation of the first order remains an integral surface when it is given an arbitrary screw motion about the \(z\) axis, prove that the equation must be of the form $$F\left(x p+y q, x q-y p, x^{2}+y^{2}\right)=0$$ If a differential equation of this type admits the quadric $$a x^{2}+b y^{2}+c z^{2}=1$$ as an integral surface, show that the characteristic curves which lie on this quadric are its intersections with the family of paraboloids \(z=k x y\).