Consider the heat equation T t JT x 0xT. Solve the equation using separation of variables (show details) under the condi- tions of the temperature being kept at zero level at the bar ends, and the initial temperature being T(x, 0)=2 sin (3x). A string with motionless ends at r = 0 and z = 1 vibrates according to the wave equation Ju t 5- U x Use separation of variables (show details) to solve the equation provided that the initial profile of the string is u(x,0) = 4 sin(2x) and the initial velocity du t It=0 = 0. Are there any points of the string, other than the ends, that do not move at all times? Explain.