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Abstract We calculate the solution of the Bagley-Torvik equation for arbitrary initial conditions and arbitrary external force as a sum of two terms. The first one is a linear combination of exponentials with error functions, and the second one is a convolution integral whose kernel is a linear combination of exponentials with error functions. The derivation of the solution is carried out by using the Laplace transform method and the calculation of a new inverse Laplace transform. The aforementioned convolution integral can be calculated for the cases of a sinusoidal or a potential-type external force. In addition, we calculate the asymptotic behaviour of the solution for t 0^+ t → 0 + and t + t → + ∞. The computation of this new analytical solution is much faster and stable than other analytical solutions found in the literature.
González‐Santander et al. (2026) studied this question.