Using Laplace Transform to solve non-linear ODE for pendulum motion and showing why it cannot be solved

Question

After solving the linear version of the ODE for the Pendulum equation using Laplace transform, I tried to use LT to solve the non-linear ODE for pendulum motion.

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However, I am not very familiar with LT and so i do not really understand how to convert the non-linear term into the Laplace domain? should I just use the sin identity, even though it doesn't look right? Once it is answered, I would also like to know why exactly it cannot be solved using LT.

Thanks in advance

Answer 1

i do not really understand how to convert the non-linear term into the Laplace domain.

Nobody else really does either. Computing the Laplace transform of even basic nonlinearities requires approximation via infinite series, see here. The only nonlinear way to combine functions that plays nice with the Laplace transform is convolution to my knowledge. The relation is given by $$ \mathcal{L}[u*v](s) = \mathcal{L}[u](s)\cdot\mathcal{L}[v](s), $$ where $$ [u*v](t) = \int_{-\infty}^\infty u(s)v(t-s)~\mathrm{d}s. $$