Laplace transform, harmonic oscillators

It should be evident that the expressions for the Laplace transforms of derivatives of functions can facilitate the solution of differential equations. A trivial example is that of the classical harmonic oscillator. Its equation of motion is given by Eq. (5-33), namely,... 

PROBLEM 2.16.6. Solve by Laplace transform methods the classical linear harmonic oscillator differential equation mdzy/cHz= —kHy(t), where kH is the Hooke s law force constant, with the initial condition dy/dt = 0 at t = 0. Note Use p for the Laplace transform variable, to not confuse it with the Hooke s law force constant kH ... 

A solution to this heat conduction problem is once again possible applying the Laplace transformation, cf. [2.18], [2.26] or [2.1], p. 317-319, where the part originating from a constant initial temperature was calculated, which fades away to zero after a sufficiently long time interval. As this solution method is fairly complicated we will choose an alternative. It can be expected that the temperature in the interior of the body also undergoes an harmonic oscillation, which with increasing depth x will be damped more and more, and in addition it will show a phase shift. The corresponding formulation for this... 

Obtain the solution for Eq. (8.55) and (8.56) for the forced harmonic oscillator using Laplace transforms. 

The Laplace transform and its inverse are well known for simple functions (Stein-feld et al., 1989 Forst, 1973). Consider, for instance, the case of a single harmonic oscillator in the classical limit, hv< kT). If the exponential in Eq. (6.47) is expanded in a power series and the higher-order terms dropped, we find that (y fclassical) = k TIhv, or using the symbol in the Laplace transform, = (3/Jv)". The density of states of a single classical harmonic oscillator is then given as follows ... 

Some differential equations can be solved by taking the Laplace transforms of the terms in the equation, applying some of the theorems presented in Section 11.3 to obtain an expression for the Laplace transform of the unknown function, and then finding the inverse transform. We illustrate this procedure with the differential equation for the less than critically damped harmonic oscillator, Eq. (12.43), which can be rewritten... 

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