How to explicitly write derivatives of a symbolic function?
I have
u = function('u',x)
and I am wondering what happens when the degrees of some scalar a
are eigenvalues of the differentiation operator (i.e. D^n u = a^n*u
). For n=1,2
there are rudimentary functional examples ( De^(a*x) = a*e^(a*x)
, sin
and cos
for a=i
and n=2
), but for higher degrees I need to go in abstract.
My question is how to symbolically assign derivatives u
? One option is to write a function that distinguishes everything ok but dispatches u
to a*u
, but what if I just want to D^3u = a^3*u
?
u
just to be "derived from
u
" (
D[...](u)(x)
), except for the third, which I want to be
a^3*u
for some scalar
a
. How could I implement this?
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What's wrong with the solution you propose in your second paragraph? e. in Maxima,
D[n](u, x) := if n=3 then a^3*u(x) else diff(u(x),x,n)$
gives you what you want, right?
Maxima allows first derivatives to be assigned symbolically with gradef
, but I don't know how to assign higher derivatives.
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