MTH108: HOMEWORK 3 – DIFFERENCE EQUATIONS I
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MTH108: HOMEWORK 3 – DIFFERENCE EQUATIONS I
Please attempt sections A and B before the tutorials.
Section A – Warm-up questions
A1. Find the four terms x1 , x2 , x3 , x4 of the solution of the following difference equations with the given initial condition:
(i) xn+1 = _3xn , x0 = 1;
(ii) xn+1 = 2xn _ 1, x0 = 3;
(iii) xn+1 = (n + xn )2 , x0 = 1;
(iv) xn+1 = 2xn _ xn 一 1 , x0 = 1, x1 = 1;
(v) xn+1 = 3xn 一 1 , x0 = 5, x1 = 1;
(vi) xn+1 = xn + 2xn 一 1 , x0 = 1, x5 = 98 .
A2. Find a constant solution to the difference equation 14xn+7 _ 9xn+5 + 2xn+1 _ 5xn 一2 = 6 .
Section B – Main questions
B1. The concentration of drug xn in a patient’s body after n days of treatment satisfies xn+1 = D+xn /(xn +3), where D > 0 is a constant determined by the daily dose of the drug . If the concentration is currently at 1 .8,
then what dose D is required to maintain the concentration at the same level?
B2. Find the solution of the following linear difference equations:
(i) xn+1 + 2xn = 3; x0 = 0 .
(ii) 3xn+2 + 8xn+1 _ 3xn = 24; x0 = 5, x1 = 1 .
(iii) xn+2 _ 2xn+1 + xn = 3n ; x0 = 1, x1 = 2 .
(iv) xn+1 _ 2xn = 3n2 _ 10; x0 = 0 .
B3. I open a bank account that pays 2% interest each month with an initial deposit of 羊 1000 . At the end of each month I make a further deposit of 羊 100 into the account . Construct a difference equation for the account balance . How long will it take for the balance to pass 羊 10000?
B4. My bank has offered me a mortgage loan of 戈 80000 at an interest rate of 3 .4% per annum to be repaid in equal monthly instalments over a period of ten years . Show that the annual interest rate is equivalent to 0 .279% per month . Write down an equation for xn+1, the balance owed at the end of month n in terms of xn and p, the fixed monthly repayment . Solve the equation and hence find the required size of the monthly repayment p .
B5. A economic model for the total national product xn in year n is given by xn+2 _ a(c + 1)xn+1 + acxn = b , for some parameters a, b, c > 0 with a 
1 . Find a particular solution . For what values of the parameters (if any) will typical solutions xn be: (i) converging to a positive constant? (ii) oscillating periodically?
Section C – Investigations
C1. Let xn be the sequence of Fibonacci numbers, that is the sequence xn satisfying xn+2 = xn+1 + xn with x0 = 0 and x1 = 1 . Show that φ = limn →& (xn+1/xn ) exists and find its value . Investigate the sequence of powers φn . Show that the values get closer and closer to integers .
C2. Investigate the so-called Tribonacci numbers, where xn+3 = xn+2 + xn+1 + xn and x0 = x1 = 0, x2 = 1 . Calculate the first 10 terms of the sequence . How fast does the sequence grow?
2023-04-21