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Let's solve this step by step manually. We are asked to verify integrability of a Pfaffian differential equation and then find its integral. The equation given is: $$ yz\,dx + (x^2y - zx)\,dy + (x^2z - xy)\,dz = 0 $$ --- ### Step (a): Verify integrability A Pfaffian differential equation of the form $$ M\,dx + N\,dy + P\,dz = 0 $$ is integrable if the following condition is satisfied: $$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}, \quad \frac{\partial M}{\partial z} = \frac{\partial P}{\partial x}, \quad \frac{\partial N}{\partial z} = \frac{\partial P}{\partial y} $$ Here, $$ M = yz, \quad N = x^2y - zx, \quad P = x^2z - xy $$ --- Step 1: Compute the partial derivatives 1. $\frac{\partial M}{\partial y} = \frac{\partial (yz)}{\partial y} = z$ $\frac{\partial N}{\partial x} = \frac{\partial (x^2y - zx)}{\partial x} = 2xy - z$ We notice immediately: _____ _______ _____ _______ _______ ____ ______ _____ ______ ____ _____ _____.
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