15. (8分) Show that the line integral Sc dy is independent of path and evaluate the given integral for any path from (2,π/6) to (0, -2).

(13) A logistic population model with relative growth rate 0.1 per year and carrying capacity 50 thousand can be expressed by the differential equation =0.1P(1 - ), with P in thousands and t in years. Given that the initial population is 9 thousand. Find the population size after 20 years. (If you memorized the formula, then you need to derive it for this problem.)

15. (8分) Show that the line integral Sc dy is independent of path and evaluate the given integral for any path from (2,π/6) to (0, -2).

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