Given the following equations: $$x'=-x+y-1$$ $$y'=-x-y+3$$ When $t=0$, $x=0$ and $y=3$ Find the general solution for $x$ and $y$ in terms of $t$.
Draw a graph of $y$ against $x$ for values of $t$ >0. Describe what happens as $t$ approaches infinity.
I have found the general solution for $x$ and $y$ to be: $$x(t)=e^{-t} (A cost+B sin t)+1$$ $$y(t)=e^{-t} (A cost+B sint)+2$$
Applying the initial conditions the particular solutions are: $$x(t)=e^{-t} (- cost+ sin t)+1$$ $$y(t)=e^{-t} ( cost+ sint)+2$$.
Can someone please help verify that this is correct so far and help me with how to proceed in graphing this and explaining the resultant graph.
Since you have explicit solutions, you can use one of the many graph plotter websites. To determine the behavior as $t\rightarrow\infty$, it might help to first look into the behaviors of $t\mapsto e^{-t}$, $t\mapsto \cos(t)$, $t\mapsto\sin(t)$ as $t$ approaches infinity and you should be able to figure a behavior for the product of those elementary functions.