Question: Method to approximate solution of ODE

Consider the following problem.

I'm looking for an implementation of the following approximation-method. Thanks to @Carl Love I already have a code for the exact solution. (It might be that my example using the given A,b,c does not have a solution but then I would just have to use different A, b, c.)


I uploaded the question as a pdf so that it is nicer to read.

I know there are probably better ways which are already part of some packages but I want to use this method.



A procedure that does the following is all I need:



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