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I was curious about what limitations the famous Laplace theorem for solving ODE had and what drawbacks it may have.

PS: I am NOT familiar with Fourier

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  • $\begingroup$ You mean the use of the Laplace transform method? It only goes forward in time, and depending on the equation it can sometimes make the problem much worse. $\endgroup$ – Ian Jan 24 '16 at 14:50
  • $\begingroup$ @Ian yes I mean: what kind of equations can't it compute, is it always the best choice to choose this method,etc,, $\endgroup$ – privetDruzia Jan 24 '16 at 14:52
  • $\begingroup$ It depends on what you mean by "can't compute". For a differential equation whose solution isn't an elementary function anyway, the Laplace transform method can tell you "the solution is the inverse Laplace transform of this horrible thing". This might be good or bad, it depends. It's also a bit awkward for BVPs. One of the nice things about the Laplace transform method for IVPs is that the initial conditions get rolled into the method of solution from the start. But with BVPs, you don't have all of the initial derivatives, so you have to keep some of them around as free parameters. $\endgroup$ – Ian Jan 24 '16 at 14:59
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    $\begingroup$ The Laplace transform method is usually of no real use in nonlinear problems, just because you don't get a nice algebraic equation out of it; one exception is that the Laplace transform of a convolution is just a product, which is convenient. Finally there are equations that we can solve and that the Laplace transform method makes much worse, such as $t^3 y'=1$. $\endgroup$ – Ian Jan 24 '16 at 14:59

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