Hi please answer question 3

also Please note that the second condition means that you first take the derivative of y'(x,t) and then plug in x = L

8207 AM Fri Nov 6 Done PCS622_HW4 (1 of 447) .3) 79% E} way, ddes this solution have the form y(.r, t) = X ($)T(t)? If so give formulas for the functions X(:r) and T(t). 3.) (4 pts) A string of length L9 is stretched between two poles. The end of string at m = 0 is tied tightly to the left pole; this end is xed. The end of the string at m = E is attached to a ring which can slide freely (without friction) up and down the right pole. The position of the tiny piece of string at position a: and time t is given by y(a:, t) which satises the wave equation 3MW0=3WQ (a where the prime ' denotes derivatives with respect to a: and the dot denotes derivatives with respect to time. The boundary conditions are MQ=0 J=0 (3 The rst condition tells us that the left end is xed. The second condition tells us that the right end is free (the m-derivative of the function must vanish at a: = H)\". We will also assume the initial conditions y(:c,0)=yi(:c) {mm = 0 (4) meaning the initial shape of the string is y(:r:) (known), and the string is released from rest. (a) By now, you know that the basis functions (eigenfunctions) for the spatial part of the wave equation Xn(:c) are trigonometric functions (sines and cosines). Draw a careful sketch of at least two trigonometric functions which satisfy Xn(0) = 0 and X71127) = D. (Hint: the rst one has wavelength 4B, the second one has wavelength 43/3) (b) Using your drawings1 can you guess the general form of the eigenfunctions Xn($) that will meet these boundary conditions? (c) Solve the wave equation using separation of variables and apply the homogeneous boundary conditions to get a general solution. You should nd Write the solution explicitly by nding X1403) and Tn(t), and by writing the possible values for 77.. Compare with your guess from part (3b)