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MAT311H5F Assignment 2 ”Due date: Saturday October 9 at 11:59 PM” To be submitted on Crowdmark

Question 1 (5 Marks). Find all functions u(x, y) that solve the equation

Question 2 (5 Marks). Find all functions u(x, y) that solve the equation

and additionally satisfy u(t, t) = −2 t for all t ∈ R.

Question 3 (3 Marks). Write the general expression for solutions u(t, x) to the following wave equation

Hint. Try to factor the equation into two first order equations like we did in class.

Question 4 (2 Marks). Find all functions u(t, x) that solve the wave equation

and additionally satisfy  for all x ∈ R.

Hint. One way to solve this question is as follows: first re-write the equation in terms of a new function v that is defined by  Choose a and b suitably so that the equation (1) becomes a standard wave equation for the function v without first order terms. Then solve the new equation for the function v and finally convert back to u.