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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?

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

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  • "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – corey979, Alex Trounev, bbgodfrey, m_goldberg, MarcoB
If this question can be reworded to fit the rules in the help center, please edit the question.

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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
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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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