Problems

In Problems 1-16, find all solutions of the differential equation.

17            A country has a population of 10 million at time t = 0 and constant annual birth rate b = 0.025 and death rate d = 0.015 per person. Find the population as a function of time.

18            Suppose a tree grows at a yearly rate equal to to of its height. If the tree is 10 ft tall now, how tall will it be in 5 years?

19            A bacteria culture is found to double in size every minute. How long will it take to increase by a factor of one million?

20            If a bacteria culture has a population of B at time t = 0 and 2B at time t = 10, what will be its population at time t = 25?

21             A city had a population of 100,000 ten years ago and its current population is 115,000. If the growth is exponential, what will its population be in 30 years?

22            A radioactive element has a half-life of 100 years. In how many years will 99% of the original material decay?

23            What is the half-life of a radioactive substance if 10 grams decay to 9 grams in one year?

24            A body of mass m moving in a straight line is slowed down by a force due to air resistance which is proportional to its velocity, F = -kv. If the velocity at time t = 0 is v₀ find its velocity as a function of time. Use Newton's law, F = ma = m dv/dt.

25            A particle is accelerated at a rate equal to its position on the y-axis, d²y/dt² = y. At time t = 0 it has position y = 2 and velocity dy/dt = 0. Find y as a function of t.

26            A mass of m grams at the end of a certain spring oscillates at the rate of one cycle every 10 seconds. How fast would a mass of 2m grams oscillate?

27            A particle is accelerated at a rate equal to its position on the j'-axis but in the opposite direction, d²y/dt² = -y. At time t = 0 it has position y = 1 and velocity dy/dt = - 2. Find y as a function of t.

28            In Problem 27, suppose that at time t = 0 the position is y = -3 and at time t = π/2 the position is y = 2. Find y as a function of f.

29            Suppose the birth rate of a country is declining so that its population satisfies a differential equation of the form dy/dt = ky/t. If y = 10,000,000 at time t = 10 and y = 20,000,000 at time t = 40, find y as a function of f.

30            Work Problem 29 under the assumption that the population satisfies a differential equation of the form dy/dt = ky/t².

31             Suppose a population satisfies the differential equation dy/dt = 10^-8y (10⁸ - y) and y₀ = 10⁷ at time t₀ = 0 years. Find the population y at time r = 1 year.

32            Suppose a population satisfies a differential equation of the form dy/dt = ky(l0⁸ - y). At time t₀ = 0 years the population is y₀ = 10⁷, and at time t₁ = 1 year the population is y₁ = 2 · 10⁷. Find y as a function of t.

33            Suppose a population grows according to the differential equation dy/dt = ky(L - y), and 0 < y < L, 0 < k.

(a)    Show that there is a single inflection point t₀, and the growth curve is concave upward when t < t₀ and concave downward when t > t₀.

(b)    Find the population y₀ at the inflection point r₀.

34            A population with a constant annual birth rate b and death rate d per person, and a constant annual immigration rate I, grows according to the differential equation dy/dt = (b - d)y + 1. Suppose b = 0.025, d = 0.015, / = 10⁴ people per year, and the population at time t = 0 is ten million people. Find the population as a function of time.

35            Suppose the population of a country has a rate of growth proportional to the difference between 10,000,000 and the population, dy/dt = k(10,000,000 - y). Find y as a function of t assuming that:

(a)    y = 4,000,000 at f = 0 and y = 7,000,000 at t = 1.

(b)    y = 13,000,000 at t = 0 and y = 11,000,000 at t = 1.

36            Find all curves with the property that the slope of the curve through each point P is equal to twice the slope of the line through P and the origin.

37            Find all curves whose slope at each point P is the reciprocal of the slope of the line through P and the origin.

38            Find all curves whose tangent line at each point (x, y) meets the x-axis at (x - 4,0).