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### Solution to PDE (i): The given PDE is: \[ (3D^2 - 2D^2 + D - 1)z = 4e^{x+y}\cos(x+y) \] First, simplify the operator: \[ 3D^2 - 2D^2 + D - 1 = D^2 + D - 1 \] So the PDE becomes: \[ (D^2 + D - 1)z = 4e^{x+y}\cos(x+y) \] #### Step 1: Solve the homogeneous equation: \[ (D^2 + D - 1)z = 0 \] The auxiliary equation is: \[ m^2 + m - _________ _________ ______ _______ ________.
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