QUESTION 6 (20 MARKS) a) Consider the wave equation π2utt​=uxx​,−∞

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QUESTION 6 (20 MARKS) a) Consider the wave equation π2utt​=uxx​,−∞<x<∞,t≥0 with u(x,0)=sinπxcosπx,ut​(x,0)=0. Find the solution by using d'Alembert's solution method. ( 5 marks) b) Consider the heat equation ∂t∂u​=4∂x2∂2u​,0≤x≤1,t≥0, with boundary conditions u(0,t)=0,ux​(1,t)=0,t≥0, and initial conditions u(x,0)=1−x,0≤x≤1. Use the method of separation of variables to determine the general solution of the equation.