In the previous post we found a solution to
using operator calculus, i.e. treating the differential operator D like a number and doing tricks with it. See the earlier post for a justification of why we can get away with unorthodox manipulations.
We can generalize the method of the previous post to say that a solution to any differential equation of the form
is given by
In the previous post we had
but the method works more generally.
We then find a power series for 1/p(D), most likely by partial fraction decomposition, and apply the result to f(x). There may be a fair amount of labor left, but it’s purely calculus; all the differential equation work is done.
Let’s use this method to find an approximate solution to
where ζ(x) is the Riemann zeta function.
In the previous post we worked out that
and so an approximate solution to our differential equation near 0 is
Numerically this works out to
If you want more terms, carry the series for 1/(D² + 7D + 12) out to more terms.