Question

Find the integral surface of quasi linear partial differential equation xzp+yzq=-xy which passes through the curve y=x²,z=x³

Answer 1

$xzp+yzq=-xy \\ \text{This is lagrange's equation}\\ Pp+Qq=R\\ \frac{dx}{p}=\frac{dy}{Q}=\frac{dz}{-xy}\\ \frac{dx}{xz}=\frac{dy}{yz} \implies \frac{dx}{x}=\frac{dy}{y}\\ \implies\int\frac{dx}{x}=\int\frac{dy}{y}\\ Inx=Iny+Ina\\ \implies In\frac{x}{y}=Ina\\ \frac{x}{y}=a\\ \frac{ydx+xdy-2zdz}{xyz+xyz-2xyz}=\frac{ydx+xdy-2zdz}{0}\\ d(xy)-2zdz=0\\ xy-z^2=b\\ \psi(\frac{x}{y},xy-z^2)=0.$