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.

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