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Prove. Given the function U(x, t) defined for a x b, t > 0. Let the Laplace transform of U(x, t) be

Then the Laplace transform of ∂U^{2}/∂t^{2} is given by

where

Proof. Let V = ∂U/∂t. Then

= s{s L[U] - U(x, 0)} - U_{t}(x, 0)

= s^{2}u - sU(x, 0) - U_{t}(x, 0)

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