Math 132: Calculus for Physical and Life Sciences 2 Problem Set 7 Due Friday, April 4, 2008, at the beginning of class.

General Directions: You must show all work for credit on these problems.

1. All parts of this question deal with the differential equation y′ = x + y.

(a) Verify that y = −x − 1 is a solution of this differential equation.

(b) Sketch the direction field of this equation by hand, showing the slopes at all points with −3 ≤ x ≤ 3, −3 ≤ y ≤ −3 with integer coordinates.

(c) On your direction field graph, sketch approximate graphs of the solutions of this differential equation satisfying the initial conditions y(0) = −2, y(0) = 0, y(0) = 1.

(d) Use Euler’s method with n = 5, Δx = .2 to approximate the solution of the equation with the initial condition y(0) = 1 for 0 ≤ x ≤ 1.

2. All parts of this question deal with the differential equation y′ = 3 − y.

(a) Sketch the direction field of this equation by hand, showing the slopes at all points with −1 ≤ x ≤ 5, 0 ≤ y ≤ 5 with integer coordinates.

(b) On your direction field graph, sketch approximate graphs of the solutions of this differential equation satisfying the initial conditions y(0) = 0, y(0) = 3, and y(0) = 4. What is the long term behavior of each?

(c) Use Euler’s method with n = 5, Δx = .2 to approximate the solution of the equation with the initial condition y(0) = 1 for 0 ≤ x ≤ 1.

(d) This is a separable equation. Derive the general solution by separating variables and integrating.

(e) What is the exact formula for the solution of the differential equation with the initial condition y(0) = 1?

3. Solve each of the following differential equations, finding an explicit formula for y as a function of x if possible (you can leave the solution in implicit form if not). In some cases you are given initial conditions; use them to determine the arbitrary constant in the general solution.

(a) y′ = \frac{tan(y)}{x^{2} + 4x + 3}, find the general solution.

(b) y′ = \frac{(x − 1)y^{5}}{x^{2}(2y^{3} − y)}, find the general solution.

(c) y′ = 2xy with y(0) = 4.

(d) y′ = x(y^{2} + 1) with y(0) = 1.

(e) y′ = 2xy^{2} + 3x^{2}y^{2} with y(1) = −1.

(f) y′ = 3y(1 − y/20) with y(0) = 5. (Note: this is a logistic equation.)

4. (Radiocarbon dating.) C^{14} is a radioactive isotope of carbon with half-life of approximately 5700 years. It occurs naturally and is incorporated in living tissues by normal metabolic processes. Assume that any C^{14} present when the tissue dies decays exponentially from that time on. Carbon extracted from an ancient skull contains only one-sixth as much C^{14} as carbon extracted from present-day bone. How old is the skull?

5. (Newton’s Law of Heating/Cooling.) A pitcher of buttermilk initially at 25^{∘} C is cooled by setting it out on the front porch on a cold day when the exterior temperature is a constant 0^{∘} C. Suppose that the buttermilk has reached 15^{∘} C after 20 minutes.

(a) What will the temperature be after 25 minutes?

(b) When will the temperature reach 5^{∘} C?

6. The population of fish in a lake is attacked by a microscopic water-borne parasite at t = 0, and as a result the population declines at a rate proportional to the square root of the population from that time on.

(a) Express this statement about the rate of growth of the population P as a differential equation. (There should be a constant of proportionality, say −k, in your equation.)

(b) Use separation of variables to find the general solution of your differential equation.

(c) At t = 0 there were 900 fish in the lake; 441 were left after 6 weeks. When did the fish population disappear entirely?