Word Count : 521

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To solve the partial differential equation (PDE): \[ \left( \frac{\partial z}{\partial x_1} \right) \left( \frac{\partial z}{\partial x_2} \right) \left( \frac{\partial z}{\partial x_3} \right) = z^2 x_1 x_2 x_3, \] we can use the method of separation of variables. Here's a step-by-step solution: ### Step 1: Assume a Separable Solution Assume that the solution \( z \) can be written as a product of functions, each depending on a single variable: \[ z(x_1, x_2, x_3) = X_1(x_1) X_2(x_2) X_3(x_3). \] ### Step 2: Compute Partial Derivatives Compute the partial derivatives of \( z \): \[ \frac{\partial z}{\partial x_1} = X_1'(x_1) X_2(x_2) X_3(x_3), \] \[ \frac{\partial z}{\partial x_2} = X_1(x_1) X_2'(x_2) X_3(x_3), \] \[ \frac{\partial z}{\partial x_3} = X_1(x_1) X_2(x_2) X_3'(x_3). \] ### Step 3: Substitute into the PDE Substitute these partial _______ ____ ___ _________ _____ ____ ___.
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