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Consider the boundary value problem  { t u t = u…

Consider the boundary value problem  { t u t = u x x + 2 u , 0 0 u ( π , t ) = 0 , t > 0 . {“version”:”1.1″,”math”:”\left\{\begin{aligned} t u_t &= u_{xx} + 2u, \quad 0 < x < \pi,\ t > 0 \\ u(0,t) &=0,\quad t > 0 \\ u(\pi,t) &= 0,\quad t > 0 .\end{aligned} \right.”} Part (a) [10 pts]: Use separation of variables to find ALL the nonzero solutions to the problem and write them in the corresponding answer box in the Solution Sheet. Show that there are an infinite number of solutions from part (a) that also satisfy the initial condition u(x,0)=0, 0

Consider the following problem for the vertical displacement…

Consider the following problem for the vertical displacement u(x,t){“version”:”1.1″,”math”:”u(x,t)”} of a string with fixed ends, given an initial displacement and struck downwards at time t=0{“version”:”1.1″,”math”:”t=0″} giving it an initial velocity:  ( ∗ ) { u t t − u x x = 0 , 0 0. {“version”:”1.1″,”math”:”u(x,t)=A\cos(\omega x)\cos(\omega t)+B\sin(\omega x)\cos(\omega t)+C\cos(\omega x)\sin(\omega t)+D\sin(\omega x)\sin(\omega t),\ \omega > 0.”} Part (a) [9 pts]: Find ALL values of ω>0{“version”:”1.1″,”math”:”ω>0″} that produce  nonzero solutions to the PDE and satisfies the two homogeneous BC and write your answer to the corresponding answer box in the Solution Sheet. Also write their corresponding “eigenfunctions” in the corresponding answer box in the Solution Sheet. Part (b) [2 pts]: Write the solution u(x,t){“version”:”1.1″,”math”:”u(x,t)”} to *{“version”:”1.1″,”math”:”*”} as a linear superposition/combination of only the functions you wrote in the second box in Part (a) above that you will use in Part (c). Part (c) [9 pts]: Apply the remaining nonhomogeneous BC to the function you wrote in the box in part (b) to find the function u(x,t){“version”:”1.1″,”math”:”u(x,t)”} that solves the full BVP. You must use the answer you wrote in the answer box in part (b) to get any credit.