please help me part a,b,c,d,e,f

The transverse displacement v = v(:1:, t) of a stretched string of length L satsifies the wave equation =c , for O0, with boundary conditions U(U,t) = 0 = v[L,t) for t > 0. Here c is a positive constant determined by the tension in, and material properties of the string. Assume that the PDE has a solution of the form v(;c, t) = F(:1:} G[t), where G' is a non-zero function. There is a constant k such that F\"kF=o and G\"c2kG=0. gyntax advice: - Use a comma to separate the functions that you found in parts (a), (b} and {e}. . Use omega for to, use Pi for a". 2 For the following question we assume that k = w where w > 0. (a) The general solution for the equation F\" kF = 0 has the form 17(3) 2 Af1(m) l sz (1:), where A and B are real constants. Find the functions f1 and f2. [f1[3)3f2($)] = ii (b) Similarly, the general solution of the equation G\" kc2G = 0 has the form Git} = 0910) + 1392 (t), where C , D are real constants. Find the functions 91 and 92. [91 (13,920)] = Q (C) Your friend Haoyu is interested in finding all non-trivial solutions F that satisfy the boundary conditions given in this question. Help your Haoyu by answering the following questions. Fm): @ FUD} = ((1) Hence, find all non-trivial solutions F. F(I)=BX @ forn=1,2,3..., where B is a constant. (e) Therefore, we can derive a sequence of functions to = vn(1:,t) for n = 1,2,3, . . . that satisfy the PDE and the boundary conditions. We have found that for each n, vn(:i:,t) can be written as a linear combination of two functions pm and q\". Find these two functions. [13458335949310] = IQ {f} Hence give the solution U(m,t) if the initial conditions are 1 10 a mam503M?) Ssm( :5\") and [z,0)=0. Enter your answer in the box below. \"(3:0 : El