Suppose I have the following third-order difference equation:
Y[t]+aY[t-1]+bY[t-2]+cY[t-3]=G d^t
I need to convert it into a differential equation and I followed this method:
y' = y(t+1)-y(t);
y'' = y'(t+1)-y'(t) = y(t+2)-2y(t+1)+y(t)
y'''=y''(t+1)-y''(t) = y(t+3)-3y(t+2)+3y(t+1)-y(t)
Rearranging the orginal expression:
y(t+3)-3y(t+2)+3y(t+1)-y(t)+(a+3)(y(t+2)-2y(t+1)+y(t))+(3-2a+b)(y(t+1)-y(t))+(1+c+b-3a)y(t) = G d^t
Which can be converted to: y'''+(a+3)y''+(3-2a+b)y'+(1+c+b-3a)y = G d^t
Is this procedure correct? How do I treat G d^t ? Thanks a lot for your help.