1. The problem is to write down the auxiliary equations used in Charpit's method for solving first-order partial differential equations. 2. Charpit's method transforms a first-order PDE of the form $$F(x,y,z,p,q)=0$$ where $$p=\frac{\partial z}{\partial x}$$ and $$q=\frac{\partial z}{\partial y}$$ into a system of ordinary differential equations called the auxiliary equations. 3. The auxiliary equations are derived from the total differential of $$F$$ and are given by: $$\frac{dx}{F_p} = \frac{dy}{F_q} = \frac{dz}{pF_p + qF_q} = -\frac{dp}{F_x + pF_z} = -\frac{dq}{F_y + qF_z}$$ where $$F_x, F_y, F_z, F_p, F_q$$ are the partial derivatives of $$F$$ with respect to $$x, y, z, p, q$$ respectively. 4. These equations form a system of characteristic ODEs that can be solved to find the solution surface of the PDE. 5. In summary, the auxiliary equations of Charpit's method are: $$\frac{dx}{F_p} = \frac{dy}{F_q} = \frac{dz}{pF_p + qF_q} = -\frac{dp}{F_x + pF_z} = -\frac{dq}{F_y + qF_z}$$ This system is fundamental in applying Charpit's method to solve nonlinear first-order PDEs.