Many countries in Europe have shown a marked decline in their growth rates over the past few decades. Below is a table for the population of France over the last part of the 20^th century.

a. A Malthusian growth model with a time-varying growth rate provides a good fit to this situation where the population growth is declining. An appropriate model is given by the following differential equation:

where t is in years after 1950. This model has a simple decreasing linear growth rate with Malthusian growth rate b. Solve this differential equation, then find the best parameters a, b, and P₀ to fit the data above. Write the general solution (with a, b, and P₀), then give the values of the parameters, which satisfy the least squares best fit parameters to the model. Give the sum of squares error after applying Excel's solver routine.

b. Graph the data and the model for 0 < t < 100 (from 1950 to 2050). What does the model predict for the population of France in 2025 and 2050? What year does the model predict that the population of France will reach its maximum and list that maximum population?

c. Describe how well the model fits the data. Give some strengths and weaknesses of this particular model.