Word Count : 577

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We are asked to solve the first-order partial differential equation: [ (p + q)(px + qy) = 1, \quad p = \frac{\partial z}{\partial x}, ; q = \frac{\partial z}{\partial y} ] using Charpit’s method. --- The given PDE can be written as: [ F(x, y, z, p, q) = (p + q)(px + qy) - 1 = 0 ] Charpit's auxiliary system is: [ \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} ] Step 1: Compute partial derivatives of (F) with respect to (p), (q), (x), and (y). [ F = (p+q)(px + qy) - 1 ] * (F_p = (px + qy) + (p+q)x = px + qy + px + qx = 2px + q(x+y)) * (F_q = (px + qy) + (p+q)y = px + qy + py + qy = p(x+y) + 2qy) ___ _____ ______ ____ ________ ____ ________ _____ ________ ___ _______ ________.
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