Question

Solve;  (𝑧 2 − 2𝑦𝑧 − 𝑦 2 )𝑝 + (𝑥𝑦 + 𝑧𝑥)𝑞 = 𝑥𝑦 − 𝑧x

Answer 1

$(z^2-2yz-y^2)z_x+(xy+zx)z_y= xy-zx\\ \text{By Charpit-Lagrange method :}\\ \frac{dx}{z2−2yz−y2}=\frac{dy}{xy+zx}=\frac{dz}{xy−zx}\\ \text{First characteristic equation,}\\ \frac{ dy}{xy+zx}=\frac{ dz}{xy−zx}\\\frac{ dz}{dy}=\frac{y−z}{y+z} \\ z^2+2yz−y^2=c1\\ \text{Second characteristic equation,}\\ \frac{xdx+ydy+zdz}{x(z^2-2yz-y^2)+y(xy+zx)+z(xy-zx)}=0\\\frac{xdx+ydy+zdz}{0}=0\\\implies xdx+ydy+zdz=0\\ x^2+y^2+z^2=c2 \\ \text{Therefore, solution is given by,}\\ x^2+y^2+z^2=f(z^2+2yz-y^2)$