Problem 1. Check stability of all equilibrium solutions of the following diference equations: (a) Xn+1 = Xn + (Xn − 1) (Xn — 4)², n = 0, 1, ...; (b) (c) Xn+1 = 2xn− x²³, n = 0, 1, ...; • 2xn — x² (2 — xn), n = 0, 1, .... - Xn+1 =

#14

3. In this exercise we use the technique of completing the square to solve an optimisation problem (without needing to use differential calculus). (a) Find real numbers k and l such that the following equation holds, where x is a real number: 1800x – 10x² = -10(x – k)? + l. (b) The number of pears produced by trees in a pear orchard depends on the density of the tree planting. It is known that if n trees are planted on one hectare of land then each tree will produce approximately 1800 – 10n pears. Write down a function P(n) of n that describes the total number of pears produced per hectare. (c) In the scenario of part (b), predict the number of trees per hectare that should be planted to maximise the pear yield and estimate the maximum number of pears yielded per hectare.

Q(1): Find the general solution of y(4) – 5y" + 4y = 10e-3".