We have a linear difference equation with constant coefficient $$\begin{cases}x(t+1)=ax(t)+by(t)\\y(t+1)=cx(t)+dy(t)\end{cases}$$
What is the differential equation associated with the above difference equation?And what is the reason of such association? What about if the coefficient are not constant?May you give a reference for more materials on such topics?Thank you.
To do this, you need to construct a derivative from the pieces you have $$\frac{dx}{dt} \approx \frac{x(t+\Delta t)-x(t)}{\Delta t}$$ For the difference equation, $\Delta t=1$, so this translates to $$\frac{x(t+1)-x(t)}{1}=\Delta x$$ So for the first equation, you need to move a $1x$ over the left side, yielding $$\frac{dx}{dt} \approx x(t+1)-x(t)=(a-1)x(t)+by(t)$$