In a certain region, about 6% of a city's population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where Subscript[x, 0] is the initial population in 2015. Then estimate the populations in the city and in the suburbs two years later, in 2017.

- Let p_c be the population in the city ( in a given year ) and p_s is the population in the suburbs ( in a given year ) . The first sentence tell us that populations p_c' and p_s' for next year would be:

0.94*p_c + 0.04*p_s = p_c'

0.06*p_c + 0.96*p_s = p_s'

- Assuming 6% moved while remaining 94% remained settled at the time of migrations.

- The matrix representation is as follows:

[tex]\left[\begin{array}{cc}0.94&0.04\\0.06&0.96\end{array}\right] \left[\begin{array}{c}p_c\\p_s\end{array}\right] = \left[\begin{array}{c}p_c'\\p_s'\end{array}\right][/tex]

- In the sequence for where x_k denotes population of kth year and x_k+1 denotes population of x_k+1 year. We have:

[tex]\left[\begin{array}{cc}0.94&0.04\\0.06&0.96\end{array}\right] x_k = x_k_+_1[/tex]

- Let x_o be the populations defined given as 10,000,000 and 800,000 respectively for city and suburbs. We will have a population x_1 as a vector for year 2016 as follows:

[tex]\left[\begin{array}{cc}0.94&0.04\\0.06&0.96\end{array}\right] x_o = x_1[/tex]

- To get the population in year 2017 we will multiply the migration matrix to the population vector x_1 in 2016 to obtain x_2.

[tex]x_2 = \left[\begin{array}{cc}0.94&0.04\\0.06&0.96\end{array}\right]\left[\begin{array}{cc}0.94&0.04\\0.06&0.96\end{array}\right] x_o[/tex]

- Where,

[tex]x_o = \left[\begin{array}{c}10,000,000\\800,000\end{array}\right][/tex]

- The population in 2017 x_2 would be:

[tex]x_2 = \left[\begin{array}{cc}0.94&0.04\\0.06&0.96\end{array}\right]\left[\begin{array}{cc}0.94&0.04\\0.06&0.96\end{array}\right] \left[\begin{array}{c}10,000,000\\800,000\end{array}\right] \\\\\\x_2 = \left[\begin{array}{c}8,920,800\\1,879,200\end{array}\right][/tex]