Assignment:
A linear PDE can be written in differential operator notation L(u) = f. where L is the linear differential operator, u is the unknown function, and f is the right-hand side function. For each of the following PDEs, determine the linear operator and the right-hand side function, the order of the PDE, and whether the PDE is homogeneous or nonhomogeneous:
(a) Uxxx + Uyyy - U = 0
(h) Utt - Uxx + Uyy + Uzz = xyz
(c) x^2Uxx - x^2Uy = cos(x) - sin(y)
(d) y^2Uxx - x^2Uy = cos(y) - sin(x)
(e) Ut - cos(xt)Uxxx - t^5 = t^2 u
Provide complete and step by step solution for the question and show calculations and use formulas.