Consider the equation

MATH 3073 1B Winter 2015 Assignment # 4 Answer the following questions. Be sure to include your workings. Be clear and explain your steps. You can discuss the assignment questions and course material with other students and with the instructor but you must write up your own solutions.  Consider the equation 3uxx + 7uxy + 2uyy = 0. (a) Verify that the equation is hyperbolic for all x and y and find the characteristic coordinates µ and η. (b) Find the new canonical equation uµη = Φ(µ, η, u, uµ, uη) for the equation 3uxx + 7uxy + 2uyy = 0. (c) A second canonical form of the general hyperbolic equation can be found by making another transformation α = µ + η, β = µ − η. Find this alternative canonical form uαα − uββ = Ψ(α, β, u, uα, uβ) for 3uxx + 7uxy + 2uyy = 0. (See Note 1 on page 181 of textbook) MATH6132 students: Also answer the following questions. 1. Describe (briefly) the Cauchy-Kowalewski Theorem. (Summarize) 2. (a) Show by differentiating the d’Alembert solution of PDE utt = c 2uxx, −∞ < x < ∞, 0 < t < ∞ ICs u(x, 0) = 0, ut(x, 0) = φ(x), −∞ < x < ∞ that you can find the solution to PDE utt = c 2uxx, −∞ < x < ∞, 0 < t < ∞ ICs u(x, 0) = φ(x), ut(x, 0) = 0, −∞ < x < ∞ (b) Apply the result from above to find the solution to PDE utt = uxx, −∞ < x < ∞, 0 < t < ∞ ICs u(x, 0) = x, ut(x, 0) = 0, −∞ < x < ∞ 3. Read Lesson 24 in textbook.