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Consider the problem of finding the temperature distribution…

Consider the problem of finding the temperature distribution u(x,t){“version”:”1.1″,”math”:”u(x,t)”} of a homogeneous one-dimensional rod of length π2{“version”:”1.1″,”math”:”π2″} with perfectly insulated ends and lateral sides, no internal heat generation and a given initial temperature profile. The boundary value problem is:  { u t − u x x = 0 , 0 0 , u x ( π 2 , t ) = 0 , t > 0 , u ( x , 0 ) = 100 sin ⁡ ( x ) , 0 0 {“version”:”1.1″,”math”:”u(x,t)=e^{-\omega^2t}\Big\{A\cos(\omega x)+B\sin(\omega x)\Big\}, \quad \omega > 0″}and  u ( x , t ) = C x + D {“version”:”1.1″,”math”:”u(x,t)=Cx+D”}where A, B, C, D{“version”:”1.1″,”math”:”A, B, C, D”} and ω{“version”:”1.1″,”math”:”ω”} are constants to be determined.  Part (a): Apply only the homogeneous BC to the form u(x,t)=Cx+D{“version”:”1.1″,”math”:”u(x,t)=Cx+D”} to find if there is a nonzero solution of this form to the completely homogeneous BVP.  Part (b): Apply only the homogeneous BC to the form  u ( x , t ) = e − ω 2 t { A cos ⁡ ( ω x ) + B sin ⁡ ( ω x ) } {“version”:”1.1″,”math”:”u(x,t)=e^{-\omega^2t}\Big\{A\cos(\omega x)+B\sin(\omega x)\Big\}”} to find if there are nonzero solutions of this form to the completely homogeneous BVP. Part (c): Write a linear superposition of only the functions listed in your answer box in part(b) to use in part(d). Part (d): Apply the one nonhomogeneous BC to your answer in part (c) to find the solution u(x,t){“version”:”1.1″,”math”:”u(x,t)”} to the full BVP. You must use the answer you wrote in part (c) above to get any credit here.