backward euler method, Exercises of Mathematics

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EXAMPLE 1 The Canadian population at ¢ = 0 (current year) is 35 million. If 2.5% of the population have a child in a given year, while the death rate is 1% of the population, what will the population be in 50 years? What if 6% of the population have a child in a given year and the death rate is kept constant at 1%, what will the population be in 50 years? SOLUTION When k = 0.015, the IVP is given by: dx oe 5 dt 0.0152 Given a value for «; at t;, and taking h = t;4; — t; = 1, the estimate for z;,, can be calculated according to the formula: wy = apt 1G =a, + A(0.015ai41) waripit=tiga Rearranging yields: + i TH = T0015 With zo = 35 million and to = 0 years, the estimate for the population at ¢ = 1 years using the implicit Euler method is given by: 35 ™=T—9015 8 at ¢ = 2 years, the value of x2 in millions is given by: 35.533, r= T0015 = 36.074 Proceeding iteratively, the following is the Excel table showing the values of the population up to ¢ = 50 years.