3. Suppose that C(x) and S(x) are two functions such that C'(x) = -S(x) and S'(x) = C(x). Prove that C2(x)+S2(x) is constant.

(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.)