1 | initial version |
In this case, the function in the question has two parameters, so it describes a vector space of dimension 2 inside the space of functions from ℝ to ℝ.
So we might look for a linear differential equation of order two.
Then for each of the functions $x \mapsto e^{-x}$ and $x \mapsto x e^{-x}$, we can look for linear relations between the function, its derivative, and its second derivative.
See Ask Sage question 56390 about that.
Then we can find a common linear dependence relation that works for both.
That will be the desired differential equation.
2 | No.2 Revision |
In this case, the function in the question has two parameters, so it describes a vector space of dimension 2 inside the space of functions from ℝ to ℝ.
So we might look for a linear differential equation of order two.
Then for each of the functions $x \mapsto e^{-x}$ and $x \mapsto x e^{-x}$, we can look for linear relations between the function, its derivative, and its second derivative.
See Ask Sage question 56390 about that.
Then we can find a common linear dependence relation that works for both.
That will be the desired differential equation.
For another approach, use power series expansions of the functions and the ore-algebra package, see in particular the guessing module.