<s>[INST] Laplace Transform to solve system of differential equations I have the system:
\begin{align}
2\frac{dx}{dt} + \frac{dy}{dt} - 2x &= 1 \\
\frac{dx}{dt} + \frac{dy}{dt} - 7x-7y &=2
\end{align}
$y(0)=0, x(0)=0$
Which I am attempting to solve using Laplace Transform's. I change the system to
$2sx+sy-2x=1$
$sx-sy-7x-7y=2$
Solving I get $x=-{(s+7)\over(s^2-9s+14)}$ and $y={3(s+1)\over(s^2-9s+14)}$.
Plugging in the inverse laplace transforms for these I get $x=\frac{14}{5}e^{7t}+\frac{9}{5}e^{2t}$ and $y=\frac{24}{5e}^{7t} - \frac{9}{5}e^{2t}$. Which is evidently wrong. 
Does anyone see what I did wrong here?
 [/INST] Using the standard Laplace transform then the equations become
\begin{align}
2(s-1) \, \overline{x} + s \overline{y} &= \frac{1}{s} \\
(s-7) \overline{x} + (s-7) \overline{y} &= \frac{2}{s}. 
\end{align}
Solving for $\overline{x}$ and $\overline{y}$ then
\begin{align}
\overline{x} &= - \frac{s+7}{s (s-2) (s-7)} \\
\overline{y} &= \frac{3 (s+1)}{s (s-2) (s-7)}
\end{align}
and lead to
\begin{align}
x(t) &= \frac{9}{5} \, e^{2 t} - \frac{2}{5} \, e^{7 t} - \frac{1}{2} \\
y(t) &= \frac{24}{35} \, e^{7 t} - \frac{9}{10} \, e^{2 t} + \frac{3}{14}.
\end{align}
Note: The Laplace transform is defined by
$$\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{- s t} f(t) \, dt,$$
in shorthand notation $f(t) \doteqdot f(s)$, and
\begin{align}
\frac{dx}{dt} &\doteqdot s \overline{x} - x(0) \\
a &\doteqdot \frac{a}{s} \hspace{5mm} \text{$a$ is a constant} \\
e^{a t} &\doteqdot \frac{1}{s - a}
\end{align} 
</s>