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Q:

Laplace transform – set of solution to differential equation

I need help with a few things.
My question is the following. Given a solution $X(t)$ to the differential equation
$$ X'(t)=aX(t)+bX(t-1) $$
I need to find the set of solutions $\{X(t)\}$ given that the Laplace Transform $X(s)$ is given by
$$ X(s)=\frac{1}{1+a(s+1)} $$
So, I solved for the general solution, taking $X(t) = \sum\limits_0^\infty{X_n(t)e^{ -st}}$ and found the recursion relations $X_n(t) = (a+n)X_n(t-1)$.
If I let $a=-1$, I get the solution
$$ X(t) = \sum\limits_0^\infty{X_n(t)e^{ -t}}$$
Do I have to find the sum of the general solution now, or could I directly write down the general solution? I’m not sure how to proceed.
And second question:
I’m trying to use the fact that $X(t)$ is a solution to the differential equation to find the solution to another differential equation.
Given
$$ X'(t)=aX(t)+bX(t-1) $$
and
$$ Y'(t)=cY(t) $$
I need to find the set of solutions $\{X(t)\}$ given that $X(t)$ is a solution to the differential equation above and $Y(t)$ is a solution to the differential equation above.
This time, I have
$$ X(s)=\frac{1}{1+a(

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