Economics Dynamics Problems 97

Economics Dynamics Problems 97 - tains carbon-14 with a...

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Continuous dynamic systems 81 2. Analyse the qualitative and quantitative properties of the Gompertz equation for population growth ˙ p = p ± ( t ) = kp ( a ln p ) 3. Solve the following separable differential equations (i) dy dx = x (1 y 2 ) 1 < y < 1 (ii) dy dx = 1 2 y + y 2 (iii) dy dx = y 2 x 2 4. Solve the following initial value problems (i) dy dx = x 2 2 x + 1 y = 1 when x = 0 (ii) dy dx = (3 x 2 + 4 x + 2) 2( y 1) y =− 1 when x = 0 5. Solve the following Bernoulli differential equations (i) dy dx y =− y 2 (ii) dy dx y = xy 2 (iii) dy dx = 2 y e x y 2 6. Show that the logistic equation dp dt = p ( a bp ) can be represented as a Bernoulli function. Using a suitable transfor- mation, solve the resulting linear differential equation; and hence show that p ( t ) = ap 0 ( a bp 0 ) e at + bp 0 where p (0) = p 0 7. In the Great Hall in Winchester hangs a round table on the wall that was purported to be King Arthur’s famous original round table top. Wood con-
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Unformatted text preview: tains carbon-14 with a decay-constant of 1.245 10 4 per year. Living wood has a rate of disintegration of 6.68 per minute per gram of sample. When the tabletop was inspected in 1977 the rate of disintegration was found to be 6.08 per minute per gram of sample. Given King Arthur was on the throne in the Ffth century AD, demonstrate that the tabletop at Winchester was not that of King Arthur. 8. (Use information in exercise 7.) In 1950 the Babylonian city of Nippur was excavated. In this excavation there was charcoal from a wooden roof beam which gave off a carbon-14 disintegration count of 4.09 per minute...
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