Can Mathematica solve infinite linear systems? For example, if we have a infinite differential system, with recurrence, is it possible to solve it with Mathematica?


closed as off-topic by corey979, Alex Trounev, bbgodfrey, m_goldberg, MarcoB Apr 23 at 12:44

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    $\begingroup$ There's no way to answer this question without a bit more detail. Can you please give an example of such a system? $\endgroup$ – Roman Apr 22 at 19:35
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    $\begingroup$ One example: $f'_n(x)=f_{n-1}(x)+Af_{n+1}(x)$. Each equation depends from two other equations in the system. $\endgroup$ – Cícero Julião Apr 22 at 19:39
  • $\begingroup$ You might be interested in RSolve. $\endgroup$ – Henrik Schumacher Apr 23 at 0:07

Mathematica can do many computations with the help of appropriate code. In your particular case, use recursion. Please try this code

ClearAll[f, x]; f[0] = 1; f[1] = x;
f[m_] := f[m] = Module[{fnp1, n = m - 1}, fnp1 /. 
      Solve[ D[f[n], x] == f[n - 1] + A*fnp1, fnp1][[1]]];

where you are free to change f[0] and f[1] to your own initial functions. Note the use of the variables x and fnp1. The fnp1 represents f[n+1] but that is not a variable according to Mathematica. Also the use of f[m_] := f[m] = to preserve the results of previous calls to f[].


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