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 = 2 z p - y^2, \quad F_q = y^2, \quad F_z = p^2, \quad F_x = 0, \quad F_y = -2 y p + 2 y q. ] Substitute into Charpit's system: [ \frac{dx}{2 z p - y^2} = \frac{dy}{y^2} = \frac{dz}{p (2 z p - y^2) + q y^2} = -\frac{dp}{0 + p p^2} = -\frac{dq}{(-2 y p + 2 y q) + q p^2}. ] Simplify one step at a time. Take: