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Charpit’s method is used to find the complete integral of a first-order partial differential equation (PDE) of the form: \[ F(x, y, z, p, q) = 0 \] where \( p = \frac{\partial z}{\partial x} \) and \( q = \frac{\partial z}{\partial y} \). ### Given PDE: \[ p^2 x + q^2 y = z \] ### Step 1: Identify Charpit’s Equations Charpit’s method introduces the characteristic equations: \[ \frac{dx}{\phi_p} = \frac{dy}{\phi_q} = \frac{dz}{p \phi_p + q \phi_q} = _______ __________ ______ __________ _____ ___ ________ ________ ________.
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