Word Count : 733

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We are given the first-order partial differential equation: [ 2xz - p x^2 - 2q xy + pq = 0, \quad p = \frac{\partial z}{\partial x}, ; q = \frac{\partial z}{\partial y}. ] Using Charpit's method, we start with the general system: [ \frac{dx}{F_p} = \frac{dy}{F_q} = \frac{dz}{pF_p + qF_q} = -\frac{dp}{F_x + p F_z} = -\frac{dq}{F_y + q F_z}. ] Here, (F(x, y, z, p, q) = 2xz - p x^2 - 2 q xy + pq). Compute partial derivatives: [ F_p = -x^2 + q, \quad F_q = -2xy + p, ] [ F_x = 2z - 2px - 2qy, \quad F_y = -2qx + 0 = -2qx, \quad F_z = 2x. ] The Charpit system is: [ \frac{dx}{-x^2 + q} = \frac{dy}{-2xy + p} = \frac{dz}{p(-x^2 + q) + q(-2xy + p)} = -\frac{dp}{(2z - 2px - 2qy) + 2x p} = -\frac{dq}{-2qx __________ ___ ________ _____ _________ ______ ___ _______ _________.
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