How would I approach a question on using d'Alembert's solution to find the solution of the initial boundary problem?

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the question i am stuck on I have obtained the following answers: c = 2 f(x) = sin(3x)cos(6t) g(x) = 28sin(7x)(cos(14*t))

however these don't appear to make sense with the boundary conditions. What is the correct way to approach a problem like this?

apologies i am new here so please bear with me as i learn the correct syntax