P9.3: The vibrations 'u,(:r, t) of a string of length L kept taut at the endpoints satisel the following PDE, called the wave equation: Where a is a physical constant that depends on the properties of the string and under the 1d and BC "(0:15): 0 = "(L3 t) "(3310) = f($):ut($10) = 9(37) (a) Separation of variables leads to the general solution u(a:, t) = Z (An cos(%t) + B sin($t)) sin(n%a:) n:0 Use the 10 to nd an expression for An, Bnl (b) Solve the heat equation 4am = uh: for a string of length 30 under the IC u(:z:, 0) = x/lO and mm, 0) = 0