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We are asked to solve the first-order partial differential equation (PDE) using Charpit’s method: $$ p^2 x + q^2 y = z, \quad \text{where } p = \frac{\partial z}{\partial x}, \, q = \frac{\partial z}{\partial y}. $$ The Charpit-Lagrange system is given by: $$ \frac{dx}{F_p} = \frac{dy}{F_q} = \frac{dz - p\,dx - q\,dy}{- (p F_p + q F_q)}. $$ Here, $F(x, y, z, p, q) = p^2 x + q^2 y - z = 0$. Then, $$ F_p = ___ _____ _____ _____ _________.
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