Word Count : 533

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The given first-order partial differential equation is: [ z p^2 - y^2 p + y^2 q = 0 ] where (p = \frac{\partial z}{\partial x}) and (q = \frac{\partial z}{\partial y}). Rewriting: [ z p^2 - y^2 p + y^2 q = 0 \implies z p^2 - y^2 p = -y^2 q \implies q = \frac{y^2 p - z p^2}{y^2} = p - \frac{z p^2}{y^2}. ] So the equation becomes: [ q = p - \frac{z p^2}{y^2}. ] Using Charpit's method, we introduce: [ F(x, y, z, p, _______ _____ _______ ________ _____ ___ _____ ____ ________ __________ ___.
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