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For the PDE $F(x,y,z,p,q)=p^{2}x+q^{2}y-z=0$ with $p=z_{x},\;q=z_{y}$, Charpit’s characteristic system is $$ \frac{dx}{F_{p}}=\frac{dy}{F_{q}}=\frac{dz}{pF_{p}+qF_{q}}=\frac{dp}{-F_{x}-pF_{z}}=\frac{dq}{-F_{y}-qF_{z}}. $$ Compute the partials: $$ F_{p}=2px,\qquad F_{q}=2qy,\qquad F_{x}=p^{2},\qquad F_{y}=q^{2},\qquad F_{z}=-1. $$ Hence the characteristic equations become $$ \frac{dx}{2px}=\frac{dy}{2qy}=\frac{dz}{2\big(p^{2}x+q^{2}y\big)}=\frac{dp}{p-p^{2}}=\frac{dq}{q-q^{2}}. $$ Since $p^{2}x+q^{2}y=z$, the third ratio _________ ____ ______ _____ ____.
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