A vibrating string fixed at x = 0 and x = L undergoes oscillations described by the wave equation 1 Pu %3D dx² c² Ət² where u(x, t) represents the displacement from equilibrium of the string. Initially the profile of the string is u(x, 0) = sin L (#) and its initial vertical velocity is du = 87 sin L It=0 (a) The four basic periodic solutions of the wave equation are A cos(kr) cos(kct) C sin(kæ) cos(kct) B cos(kr) sin(kct) D sin(kx) sin(kct) where k is a real constant and A, B,C and D are arbitrary constants. Use the boundary conditions to find the allowed values for the separation constant k. (b) Obtain the solution u(x,t) of the wave equation that also satisfies the initial conditions.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.