Solve using charpit auxiliary equations
$p=$ partial derivative of $z$ w.r.t. $x$
$q=$ partial derivative of $z$ w.r.t. $y$
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1your thoughts, ideas or efforts? – Kushal Bhuyan Dec 15 '15 at 10:48
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Using Charpit , I think you'll get only one solution, and I doubt that this has three different complete integrals. – Kushal Bhuyan Dec 15 '15 at 13:08
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See https://math.stackexchange.com/questions/3409542/the-equation-pq-px-qy-has-more-than-one-complete-integral/4863016#4863016. – Gonçalo Feb 21 '24 at 21:12