Question

Solve
px^2+2qxy-pq-2xz=0

Answer 1

$$f(x, y,z,p,q)=px^2+2qxy-pq-2xz=0$$

We have the system of Charpit’s auxiliary equations as

$$\dfrac{dx}{-f_p}=\dfrac{dy}{-f_q}=\dfrac{dz}{-pf_p-qf_q}=\dfrac{dp}{f_x+pf_z}=\dfrac{dq}{f_y+qf_z}$$

$$\dfrac{dx}{q-x^2}=\dfrac{dy}{p-2xy}=\dfrac{dz}{-px^2+pq-2qxy+pq}=$$

$$=\dfrac{dp}{2px+2qy-2z-2px}=\dfrac{dq}{2qx-2qx}$$

Now $dq=0$ or $q=a$

$$px^2+2axy-pa-2xz=0$$

$$p(x^2-a)=2x(z-ay)$$

$$p=\dfrac{2x(z-ay)}{x^2-a}$$

$$dz=pdx+qdy$$

$$dz=\dfrac{2x(z-ay)}{x^2-a}dx+ady$$

$$\dfrac{dz-ady}{z-ay}=\dfrac{2x}{x^2-a}dx$$

Integrate both sides

$$\ln(|z-ay|)=\ln(|x^2-a|)+\ln c$$

$$z=c(x^2-a)+ay$$