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Equation of Motion

The equation of motion of a particle moving under the influence of a linear restoring force and fluid friction is given by,

where, v = dq/dt. We are assuming that system has only one degree of freedom, i.e. as the particle is displaced, only one of its co-ordinate q changes. The equation is rewritten as,

let the solution of above equation be,

Substituting, we get

This is the characteristic equation and its root are

Hence, the two independent solutions are,

The general solution is the linear superposition of the above solutions, i.e.