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Math · Calculus
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1. A population of fish has a growth rate proportional to the amount of fish present at that time, with a proportionality factor of ^ per unit time (a) Write a differential equation of the form P, F(P), which models this situation, where P is the number of fish as a function of time (b) Now, assume that we have the same fish population, reproducing as above, but we are harvesting fish at a constant rate of 100 fish per unit time. Write the differential equation in this case. (c) Assume that the initial fish population is 600 fish. Solve the ordinary differential equation in part (b) above with this given initial condition Answer for (c): P(t) 500 + 100et/5 2. Find the general solution to the following ODE y,-Sy2 cos(t)-7y2 sin (4t) = 0 7 Answer: )(-8sin(t) + cos(4t) c 3. A radioactive material has a decay rate proportional to the amount of radioactive material present at that time, with a proportionality factor of 2 per unit time (a) Write a differential equation of the form P, F(P), which models this situation, where P is the amount of radioactive material (measured in micrograms) as a function of time (b) Now, assume that we have the same radioactive material decaying as above, but we are adding additional material (of the same type) at a constant rate of 6 micrograms per unit time. Write the differential equation in this case (c) Solve the ordinary differential equation in part (b) above, assuming the initial amount of radioac- tive material is 70 micrograms

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