Math 124B/215B – Assignment 1
澳洲网课代考推荐 Using separation of variables, write down the series expansion for the general solution to the damped wave equation
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(1) Using separation of variables, write down the series expansion for the general solution to the damped wave equation with Dirichlet boundary conditions
utt(x, t) + ut(x, t) − uxx(x, t) = 0, 0 < x < 1, −∞ < t < ∞
u(0, t) = u(1, t) = 0, −∞ < t < ∞.
(2) Find all nonzero solutions X(x), λ of the problem
− X 00(x) = λX(x), 0 ≤ x <≤ <a href="https://paperdaixie.com/澳洲网课代考推荐/">澳洲网课代考推荐</a>) = 0.
X(0) = 0, X0(
(3) Show that the problem
− X00(x) = λX(x), 0 < x < X(0) = 0, X0() + X(`) = 0
has no nonzero solutions X(x) when λ ≤ 0. (You don’t have to find any solutions.)
(4) Define the function
(a) Find the Fourier sine series expansion of f(x). You may use the formula
(b) Find the Fourier cosine series expansion of f(x). You may use the formula
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