Laplace transforms of derivatives

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,... 

Equation 3.83 can be used to determine Laplace transforms of derivatives provided that the derivatives satisfy the appropriate conditions of continuity and boundedness as required by Theorem 3.8. For example... 

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