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Given the problem utt - c²Δu = 0, x in R³, t > 0 u(x, 0) = f(x), ut(x, 0) = g(x), x in R³. (a) If F: R → R is a C² function, find a condition on α₁, α₂, α₃ so that u(x₁, x₂, x₃, t) = F(α₁ x₁ + α₂ x₂ + α₃ x₃ - t) is a solution of the equation in (P₁). (b) Find a relationship between f and g so that the plane wave solution F(α₁ x₁ + α₂ x₂ + α₃ x₃ - t) is a solution of (P₂). (c) Find all plane wave solutions of (P₁) which satisfy u(x₁, x₂, x₃, 0) = x₁ - x₂ + 1
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