Differential Equations - exercises
1.) Test: Is the given function y the general solution of the indicated differential equation?
a.) Equation: y = 2x - y; Function: y = Ae-x + 4x - 2;
b.) Equation: y = 2y - ex; Function: y = ex + Ae2x;
c.) Equation: xy = 2y Function: y = Ax2
2.) For each of these separable differential equations, find the general solution and then find the particular solution satisfying the
initial condition. Sketch the graph of your solution.
a.) y = 3x2y; y(0) = 1/2
b.) y = 3x(1/2); y(0) = 7
c.) y = x(8 - y); y(0) = 5
d.) y = 7 + 4y; y(0) = 6
dy
e.) y = xy + x; y(2) = 13.78
f.) dt = .01(25 -y) y(0) = 40
3.) Under ideal conditions, the growth of a bacterial culture will be proportional to the size at any time t (N (t) = k N for some
constant k). Consider such an ideal culture which is size 25 g at time 3 hours and size 40 g at time 5 hours.
a.) Give the function N(t) giving size at every time t
b.) Sketch the graph of your function
c.) When does the culture reach size 50 g? size 100 g?
d.) What is the growth rate of the culture (g/hr) when the size is 50 g? when it is 100 g?
For 4-, a.) Write the differential equation of the growth law involved, b.) Find the growth function N= (the particular solution. in
each case) c.) Sketch the graph of your solution d.) Answer any questions attached. [These all use the three growth models ideal, limited space, logistic - discussed in class]
4.) In a bacterial population, the growth rate is proportional to the population size, with constant of proportionality .001. The
population size at time 0 is 150g,. d.) Give both population size and growth rate at t=43 and at t=8.
5.) A bacterial population grows so that the growth rate is proportional to population size at each time t. The population starts at 25
g and has doubled after 10 hours. d.) What will be the size after 12 hours? What is the growth rate at 12 hours?
6.) The concentration D (mol/l) of a drug in the bloodstream is decreasing at a rate which is always proportional to the
concentration. The initial concentration is .15 mol/l and after two hours the concentration has decreased to .08 mol/l. d.) When
will the concentration decrease to .06 mol/l ? What will be the rate of change of the concentration at that time?
7.) A population of birds on a small island grows so that the annual growth rate is proportional to the difference between the
population N and the carrying capacity 3500 birds, with constant of proportionality .015. The population size is 1000 birds at
time 0. d.)Give the population size, the unused capacity and the growth rate at t= 5 years and t=8years
8.) With the same natural growth as in #4 (growth rate proportional to 3500 - N, with constant of proportionality .015), a sudden
immigration of birds of the same species (driven out of other nesting grounds by human expansion) raises the population to
5000 birds. Take the time of this immigration as time 0, and do steps b.) and c.) d.) Assuming this model applies, how large
will the population be ten years after the immigration (t=10)?
9.)Consider a culture of Escherichia coli growing in an environment in which the upper limit on the number of cells is 6X10 10 cells,
with growth rate (cells/hr) proportional to the difference 6X1010 - N, with constant of proportionality 4X10-3. The culture
starts (t=0 hr) at size 3X107 cells.
10.) In a population of fruit flies in a laboratory, the growth rate is jointly proportional to the population size N and to the space
remaining for growth (capacity of the environment is 625 flies), with constant .0015. The population starts at 50 flies. d.)
When is the growth rate a maximum? more of d.)Give the population size, the unused capacity, and the growth rate of the
population at t=3 days and at t=7 days.
dN
11.) The growth rate of a population of termites satisfies the equation dt = .0001(650-N)N, and the size at time 0 is 100. d.) When
is the population growing most rapidly?
12.) An epidemic of flu spreads through a population of 10,000 people. At each time t days, the rate of spread (rate of growth of the
number of people who have or have had the disease - N') is jointly proportional to the number of people infected N and to the
number of people as yet uninfected (10000-N), with constant of proportionality .00004. The epidemic begins with 10 infected
people. When will 5,000 people be infected?