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Math Help - linear difference eqn-scary exams

  1. #1
    n22
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    linear difference eqn-scary exams

    Dear Mathematicians,
    I have a question for you.thanks.
    Lets say that you had to use a linear difference equation to solve some population problem.
    How do you determine how many orders it is ?ie.first order,second order,3rd order etc.
    eg.so how would I determine (if I am finding for n+1 or not)the long term chance of survival after contracting a deadly virus.
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  2. #2
    MHF Contributor

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    Re: linear difference eqn-scary exams

    A linear difference equation can be written in terms of "differences", \Delta^n a(i) ( \Delta a(i)=  a(i+1)- a(i), \Delta^2 a(i)= \Delta a(i+1)- \Delta a(i), etc.) or in terms of different subscripts: a(i+ n), a(i+ n- 1), etc.. The "order" of a difference equation is, in the first case, the largest power of \Delta and, in the second, the largest "n".

    For example \Delta^3 a(n)- 3\Delta^2 a(n)+ 2\Delta a(n)- a(n)= 0 is a third order equation because of the \Delta^3.
    Since \Delta a(n)= a(n+1)- a(n), \Delta^2(x)= \Delta (a(n+1)- a(n))= a(n+2)- a(n+1)- (a(n+1)- a(n))= a(n+2)- 2a(n+1)+ a(n), and \Delta^3 a(n)= \Delta (a(n+2)- 2a(n+1)+ a(n))= (a(n+3)- 2a(n+2)+ a(n+1))- (a(n+2)- 2a(n+1)+ a(n))= a(n+3)- 3a(n+2)+ 3a(n+1)- a(n) that same equation can be written as a(n+3)- 6a(n+2)+ 11a(n+1)-4a(n)= 0. That, again, has order 3 because of the "n+3".

    eg.so how would I determine (if I am finding for n+1 or not)the long term chance of survival after contracting a deadly virus.
    I don't understand the "e.g.". Knowing the order does not greatly help in solving the equation. There are a large variety of ways of solving such equations.
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