(a) Use partial fractions to show that % du u2 a2 = 1 2a ln ____ u a u + a ____ + C (b)

In Problems 18, solve each pure-time differential equation. dy dx = x + sin x, where y0 = 0 for x0 = 0

In Problems 18, solve each pure-time differential equation. dy dx = e3x , where y0 = 10 for x0 = 0

In Problems 18, solve each pure-time differential equation. dy dx = 1 x , where y0 = 0 when x0 = 1

In Problems 18, solve each pure-time differential equation. dy dx = 1 1 + x2 , where y0 = 1 when x0 = 0

In Problems 18, solve each pure-time differential equation. dx dt = 1 1 t , where x(0) = 2

In Problems 18, solve each pure-time differential equation. dx dt = cos(2(t 3)), where x(3) = 1

In Problems 18, solve each pure-time differential equation. ds dt = _ 3t + 1, where s(0) = 1

In Problems 18, solve each pure-time differential equation. dh dt = 5 16t2, where h(3) = 11

Suppose that the volume V(t) of a cell at time t changes according to dV dt = 1 + cos t with V(0) = 5 Find V(t).

Suppose that the amount of phosphorus in a lake at time t, denoted by P(t), follows the equation dP dt = 3t + 1 with P(0) = 0 Find the amount of phosphorus at time t = 10.

In Problems 1116, solve the given autonomous differential equations. dy dx = 3y, where y0 = 2 for x0 = 0

In Problems 1116, solve the given autonomous differential equations. dy dx = 2(1 y), where y0 = 2 for x0 = 0

In Problems 1116, solve the given autonomous differential equations. dx dt = 2x, where x(1) = 5

In Problems 1116, solve the given autonomous differential equations. dx dt = 1 3x, where x(1) = 2

In Problems 1116, solve the given autonomous differential equations. dh ds = 2h + 1, where h(0) = 4

In Problems 1116, solve the given autonomous differential equations. dN dt = 5 N, where N(2) = 3

Suppose that a population, whose size at time t is denoted by N(t), grows according to dN dt = 0.3N(t) with N(0) = 20 Solve this differential equation, and find the size of the population at time t = 5.

Suppose that you follow the size of a population over time. When you plot the size of the population versus time on a semilog plot (i.e., the horizontal axis, representing time, is on a linear scale, whereas the vertical axis, representing the size of the population, is on a logarithmic scale), you find that your data fit a straight line which intercepts the vertical axis at 1 (on the log scale) and has slope 0.43. Find a differential equation that relates the growth rate of the population at time t to the size of the population at time t.

Suppose that a population, whose size at time t is denoted by N(t), grows according to 1 N dN dt = r (8.41) where r is a constant. (a) Solve (8.41). (b) Transform your solution in (a) appropriately so that the resulting graph is a straight line. How can you determine the constant r from your graph? (c) Suppose now that, over time, you followed a population which evolved according to (8.41). Describe how you would determine r from your data.

Assume thatW(t) denotes the amount of radioactive material in a substance at time t. Radioactive decay is then described by the differential equation dW dt = W(t) with W(0) = W0 (8.42) where is a positive constant called the . (a) Solve (8.42). (b) Assume that W(0) = 123 gr and W(5) = 20 gr and that time is measured in minutes. Find the decay constant and determine the half-life of the radioactive substance.

Suppose that a population, whose size at time t is given by N(t), grows according to dN dt = 1 100 N2, with N(0) = 10 (8.43) (a) Solve (8.43). (b) Graph N(t) as a function of t for 0 t < 10. What happens as t 10? Explain in words what this means.

Denote by L(t) the length of a fish at time t, and assume that the fish grows according to the von Bertalanffy equation dL dt = k(34 L(t)) with L(0) = 2 (8.44) (a) Solve (8.44). (b) Use your solution in (a) to determine k under the assumption that L(4) = 10. Sketch the graph of L(t) for this value of k. (c) Find the length of the fish when t = 10. (d) Find the asymptotic length of the fish; that is, find limt L(t).

Denote by L(t) the length of a certain fish at time t, and assume that this fish grows according to the von Bertalanffy equation dL dt = k(L L(t)) with L(0) = 1 (8.45) where k and L are positive constants. A study showed that the asymptotic length is equal to 123 in and that it takes this fish 27 months to reach half its asymptotic length. (a) Use this information to determine the constants k and L in (8.45). [: Solve (8.45).] (b) Determine the length of the fish after 10 months. (c) How long will it take until the fish reaches 90% of its asymptotic length?

Let N(t) denote the size of a population at time t. Assume that the population exhibits exponential growth. (a) If you plot log N(t) versus t, what kind of graph do you get? (b) Find a differential equation that describes the growth of this population and sketch possible solution curves.

Use the partial-fraction method to solve dy dx = y(1 + y) where y0 = 2 for x0 = 0.

Use the partial-fraction method to solve dy dx = y(1 y) where y0 = 2 for x0 = 0.

Use the partial-fraction method to solve dy dx = y(y 5) where y0 = 1 for x0 = 0.

Use the partial-fraction method to solve dy dx = (y 1)(y 2) where y0 = 0 for x0 = 0.

Use the partial-fraction method to solve dy dx = 2y(3 y) where y0 = 5 for x0 = 1.

Use the partial-fraction method to solve dy dt = 1 2 y2 2y where y0 = 3 for t0 = 0.

In Problems 3134, solve the given differential equations dy dx = y(1 + y)

In Problems 3134, solve the given differential equations dy dx = (1 + y)2

In Problems 3134, solve the given differential equations dy dx = (1 + y)3

In Problems 3134, solve the given differential equations dy dx = (3 y)(2 + y)

(a) Use partial fractions to show that % du u2 a2 = 1 2a ln ____ u a u + a ____ + C (b) Use your result in (a) to find a solution of dy dx = y2 4 that passes through (i) (0, 0), (ii) (0, 2), and (iii) (0, 4).

Find a solution of dy dx = y2 + 4 that passes through (0, 2).

Suppose that the size of a population at time t is denoted by N(t) and that N(t) satisfies the differential equation dN dt = 0.34N _ 1 N 200 _ with N(0) = 50 Solve this differential equation, and determine the size of the population in the long run; that is, find limt N(t).

Assume that the size of a population, denoted by N(t), evolves according to the logistic equation. Find the intrinsic rate of growth if the carrying capacity is 100, N(0) = 10, and N(1) = 20.

Suppose that N(t) denotes the size of a population at time t and that dN dt = 1.5N _ 1 N 50 _ (a) Solve this differential equation when N(0) = 10. (b) Solve this differential equation when N(0) = 90. (c) Graph your solutions in (a) and (b) in the same coordinate system. (d) Find limt N(t) for your solutions in (a) and (b).

Suppose that the size of a population, denoted by N(t), satisfies dN dt = 0.7N _ 1 N 35 _ (8.46) (a) Determine all equilibria by solving dN/dt = 0. (b) Solve (8.46) for (i) N(0) = 10, (ii) N(0) = 35, (iii) N(0) = 50, and (iv) N(0) = 0. Find limt N(t) for each of the four initial conditions. (c) Compare your answer in (a) with the limiting values you found in (b).

Let N(t) denote the size of a population at time t. Assume that the population evolves according to the logistic equation. Assume also that the intrinsic growth rate is 5 and that the carrying capacity is 30. (a) Find a differential equation that describes the growth of this population. (b) Without solving the differential equation in (a), sketch solution curves of N(t) as a function of t when (i) N(0) = 10, (ii) N(0) = 20, and (iii) N(0) = 40.

Logistic growth is described by the differential equation dN dt = rN _ 1 N K _ The solution of this differential equation with initial condition N(0) = N0 is given by N(t) = K 1 + ( K N0 1)ert (8.47) (a) Show that r = 1 t ln _ K N0 N0 _ + 1 t ln _ N(t) K N(t) _ (8.48) by solving (8.47) for r . (b) Equation (8.48) can be used to estimate r . Suppose we follow a population that grows according to the logistic equation and find that N(0) = 10, N(5) = 22, N(100) = 30, and N(200) = 30. Estimate r .

Selection at a Single Locus We consider one locus with two alleles, A1 and A2, in a randomly mating diploid population. That is, each individual in the population is either of type A1A1, A1A2, or A2A2.We denote by p(t) the frequency of the A1 allele and by q(t) the frequency of the A2 allele in the population at time t. Note that p(t)+q(t) = 1.We denote the fitness of the Ai Aj type by wi j and assume that w11 = 1, w12 = 1s/2, and w22 = 1s, where s is a nonnegative constant less than or equal to 1. That is, the fitness of the heterozygote A1A2 is halfway between the fitness of the two homozygotes, and the type A1A1 is the fittest. If s is small, we can show that, approximately, dp dt = 1 2 sp(1 p) with p(0) = p0 (8.49) (a) Use separation of variables and partial fractions to find the solution of (8.49). (b) Suppose p0 = 0.1 and s = 0.01; how long will take until p(t) = 0.5? (c) Find limt p(t). Explain in words what this limit means.

In Problems 4452, solve each differential equation with the given initial condition. dy dx = 2 y x , with y0 = 1 if x0 = 1

In Problems 4452, solve each differential equation with the given initial condition. dy dx = x + 1 y , with y0 = 2 if x0 = 0

In Problems 4452, solve each differential equation with the given initial condition. dy dx = y x + 1 , with y0 = 1 if x0 = 0

In Problems 4452, solve each differential equation with the given initial condition. dy dx = (y + 1)ex , with y0 = 2 if x0 = 0

In Problems 4452, solve each differential equation with the given initial condition. dy dx = x2 y2, with y0 = 1 if x0 = 1

In Problems 4452, solve each differential equation with the given initial condition. dy dx = y + 1 x 1 , with y0 = 5 if x0 = 2

In Problems 4452, solve each differential equation with the given initial condition. du dt = sin t u2 + 1 , with u0 = 3 if t0 = 0

In Problems 4452, solve each differential equation with the given initial condition. dr dt = ret , with r0 = 1 if t0 = 0

In Problems 4452, solve each differential equation with the given initial condition. dx dy = 1 2 x y , with x0 = 2 if y0 = 3

In a case study by Taylor et al. (1980) in which the maximal rate of oxygen consumption (in ml s1) for nine species of wild African mammals was plotted against body mass (in kg) on a loglog plot, it was found that the data points fall on a straight line with slope approximately equal to 0.8. Find a differential equation that relates maximal oxygen consumption to body mass.

Consider the following differential equation, which is important in population genetics: a(x)g(x) 1 2 d dx [b(x)g(x)] = 0 Here, b(x) > 0. (a) Define y = b(x)g(x), and show that y satisfies a(x) b(x) y 1 2 dy dx = 0 (8.50) (b) Separate variables in (8.50), and show that if y > 0, then y = C exp _ 2 % a(x) b(x) dx _

When phosphorus content in was plotted against phosphorus content of its algal food on a loglog plot, a straight line with slope 1/7.7 resulted. (See Sterner and Elser, 2002; data from DeMott et al., 1998.) Find a differential equation that relates the phosphorus content of to the phosphorus content of its algal food.

This problem addresses Malthuss concerns. Assume that a population size grows exponentially according to N(t) = 1000et and the food supply grows linearly according to F(t) = 3t (a) Write a differential equation for each of N(t) and F(t). (b) What assumptions do you need to make to be able to compare whether and, if so, when food supply will be insufficient? Does exponential growth eventually overtake linear growth? Explain. (c) Do aWeb search to determine whether food supply has grown linearly, as claimed by Malthus.

At the beginning of this section, we modified the exponentialgrowth equation to include oscillations in the per capita growth rate. Solve the differential equation we obtained, namely, dN dt = 2 (1 + sin(2t)) N(t) with N(0) = 5.