Solving system of 1st order ODE's using Laplace transforms - stuck on algebra

Question

Here is the system:

$$x'(t) + y'(t) + x(t) + y(t) = 1$$ $$y'(t) - x(t) + y(t) = -t$$

I simplified this to the following system of simultaneous equations, with ${\scr L}[x] = X$ and ${\scr L}[y] = Y$

$$X(s+1) + Y(s+1) = \frac{1}{s}$$ $$-X+Y(s+1) = \frac{-1}{s^2}$$

But I'm not sure how to solve these for $X,Y$ with the presence of $s+1$ everywhere.

Working would be great.

Thank you

Answer 1

I think you also should have some initial values $x(0)$ and $y(0)$, but maybe they are zero?

If the equations you got are correct, then, by looking at them, I suggest that you subtract the second from the first, to get $$ \bigl((s+1)X+(s+1)Y\bigr)-\bigl(-X+(s+1)Y\bigr)=\frac{1}{s}+\frac{1}{s^2}. $$ Simplifying the left-hand side, $$ (s+2)X=\frac{1}{s}+\frac{1}{s^2}. $$ Once you have solved for $X$, insert it into one of the original equations, and you will get $Y$.

I guess you can finish now?