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The problem requires finding the deflection \( y(x,t) \) of a vibrating string of unit length, fixed at both ends, with zero initial deflection and a given initial velocity \( u(x) \). The wave equation governing the motion is: \[ \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} \] where \( c \) is the wave speed. The boundary conditions are: \[ y(0,t) = 0, \quad y(1,t) = 0 \] ### Step 1: Using D'Alembert's Solution The general solution ________ ________ ________ ______ ___ ____ _________ _______ ______.
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