I am asked to Give an example of a linear system for which $(e^{−t} , \alpha)$ is a solution for every constant $\alpha$ and to give the general solution for this system. I came up with a linear system based on my understanding,but it seems wrong to me, and I would like some guidance on how to approach this type of problem
From what I understand we can broadly describe a linear system of differential equations as $$x_1' = f(t,x_1,x_2)\\ x_2' = f(t,x_1, x_2)$$
and so since I am given the solution $(e^{-t}, \alpha)=(x_1(t), x_2(t))$, A linear system for which this is a solution can be given by
$$x_1' = -e^{-t}\\ x_2' = 0$$
but to me this appears too simple and incorrect. If so how do I go about approaching such a problem?