Math 124B/215B – Assignment 1

Due:  04/12/20 18h00

MATH数学代写 Label your problems clearly, and remember to the select page  number (or numbers)  for each part of each problem which you upload to Gradescope.

Instructions: Submit your assignment to Gradescope as you did for the practice assign- ment. Label your problems clearly, and remember to the select page  number (or numbers)  for each part of each problem which you upload to Gradescope. Your solutions will be graded for mathematical clarity and correctness.

(1)Usingseparation of variables, write down the series expansion for the general solution to the damped wave equation with Dirichlet boundary conditions  MATH数学代写

u_tt(x, t) + u_t(x, t) − u_xx(x, t) = 0, 0 < x < 1, −∞ < t < ∞

u(0, t) = u(1, t) = 0, −∞ < t < ∞.

(2)Findall nonzero solutions X(x), λ of the problem

− X^jj(x)  =  λX(x), 0 ≤ x <≤ A X(0) = 0, X^j(A) = 0.  MATH数学代写

(3)Show that theproblem

−  X^jj(x) =  λX(x), 0 < x < A X(0) = 0, X^j(A) + X(A) = 0

has no nonzero solutions X(x) when λ ≤ 0. (You don’t have to find any solutions.)

(4)Define thefunction  MATH数学代写

(a)Find the Fourier sine series expansion of f (x). You may use theformula

∫ x sin αx dx = − 1 x cos αx + 1 sin αx + C.   MATH数学代写

(b)Find the Fourier cosine series expansion of f (x). You may use theformula

∫ x cos αx dx = 1 x sin αx + 1 cos αx + C.