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, q) = z p^2 - y^2 p + y^2 q = 0. ] Charpit's auxiliary equations are: [ \frac{dx}{F_p} = \frac{dy}{F_q} = \frac{dz}{p F_p + q F_q} = -\frac{dp}{F_x + p F_z} = -\frac{dq}{F_y + q F_z}. ] First, compute the partial derivatives: [ F_p ____ ________ __________ ___ ___ __________ ___ ___ ___.
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