Question

The singular integral of z=xp+yq-logpq

Answer 1

The complete integral of partial differential equation $z=xp+yq- logpq$

Let $F=px+qy+p^2+q^2=0$ . Then by Charpit's auxiliary equations

$dpFx+pFz=dqFy+qFz=dz−pFp−qFq=dx−Fp=dy−Fq$

we have

$dp0=dq0=dz−p(x+2p)−q(y+2q)=dx−(x+2p)=dy−(y+2q) .$

This implies $p=a=constant$ . Putting this in given equation, $q=−y±y2−4(a2+ax−z)$