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I've spent some weeks now trying to learn how to solve ordinary differential equations, and I am now studying the Laplace transform and how this can be applied to solve ODEs.

I feel a little bit frustrated, because it seems like I did not really need to know all the other methods, I can just use the Laplace transform all the time. Is this true? Is there no point in learning the other methods? (I'm really hoping the answer is that I am wrong about this).

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Of course not. Actually most differential equations can't be solved neither by quadratures (Google Liouville theory analogue of Galois theory) nor by other methods. It is also very easy to produce an equation for which we can't describe phase space. The only partially complete theory exists for Linear differential equations and more particularly for Linear differential equations with constant coefficients which are in essence branch of Linear Algebra. You were using Laplace transform to solve probably second order ODE with constant coefficients. Teaching kids how to "solve" ODE is IMHO fundamentally wrong approach in teaching ODEs which sets them to the wrong path for life but that is discussion for my colleges and me not the students.

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  • $\begingroup$ Thank you, Punosevac! I needed to hear this! $\endgroup$
    – Akitirija
    Commented Jan 25, 2015 at 20:10

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