Non-Zero Solutions: Algorithm to Determine Existence

[mathmari]

Hey!

I have concluded to the following results: 1. An homogeneous linear differential equation in the ring $\mathbb{C}[x]$ has a solution if at least one root of the characteristic equation is equal to $0$. 2. An homogeneous linear diffeential equation in the ring $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ has always a solution.
3. An homogeneous linear differential equation in the ring $\mathbb{C}[x, e^{\lambda x} \mid \lambda \in \mathbb{C}]$ has always a solution. So, is there an algorithm that, given an equation $Dy=0$ and inequations $D_i y\neq 0$, determines whether the system $\displaystyle{Dy=0 \wedge D_i y\ne 0}$ has non-zero solutions in the above rings?

 

[gtoguy97]

Yes, there is an algorithm that can be used to determine if the system $Dy=0 \wedge D_i y\ne 0$ has non-zero solutions in the above rings. This algorithm involves first finding the roots of the characteristic equation and then checking if any of the roots are equal to zero. If any of the roots are equal to zero, then there is a solution for the differential equation. Additionally, once all of the roots have been determined, the algorithm can also be used to check if any of the inequations $D_i y \neq 0$ are satisfied by the solutions.