Back{Use of Tech} Analytical solution of the predator-prey equations The solution of the predator-prey equations
X'(t) = -ax + bxy,y’(t) = cy - dxy
can be viewed as parametric equations that describe the solution curves. Assume a, b, c, and d are positive constants and consider solutions in the first quadrant.
a. Recalling that dy/dx = y(t)/x′(t), divide the first equation by the second equation to obtain a separable differential equation in terms of x and y.
Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min.
c. At what time does the mass of sugar reach 95% of its steady-state level?
Blowup in finite time Consider the initial value problem y'(t) = yⁿ + 1, y(0) = y₀, where n is a positive integer.
b. Solve the initial value problem with n = 2 and y₀ = 1/√2.
45–48. General first-order linear equations Consider the general first-order linear equation y'(t)+a(t)y(t)=f(t). This equation can be solved, in principle, by defining the integrating factor p(t)=exp(∫a(t)dt). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes
p(t)(y′(t) + a(t)y(t)) = d/dt(p(t)y(t)) = p(t)f(t).
Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor.
y′(t) + (2t)/(t² + 1)y(t) = 1 + 3t², y(1) = 4
42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.
b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.)
y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1
{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation
dP/dt=kP(1−P/A),P0=P_0,
where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.
c. For a fixed value of K and A, describe the long-term behavior of the solutions, for any P0 with 0<P0<A.
{Use of Tech} Logistic equation for spread of rumors Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0≤y≤1, knows the rumor, while the remaining fraction 1−y does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to y1−y. Therefore, the equation that describes the spread of the rumor is y′(t)=ky(1−y), for t≥0 where k is a positive real number and t is measured in weeks. The number of people who initially know the rumor is y(0)=y0, where 0≤y0≤1.
a. Solve this initial value problem and give the solution in terms of k and y0.
{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation
m · v'(t) = mg + f(v)
mass | acceleration | external forces
where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.
c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a.
{Use of Tech} Free fall Using th e background given in Exercise 47, assume the resistance is given by f(v)=−Rv, for t≥0, where R>0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil).
c. Find the solution of this separable equation assuming v(0)=0 and 0<v<g/b.
{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm.
a. Find the solution of the initial value problem.
{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm.
b. Find the solution in k=0.1the case that and H=0.5m.
{Use of Tech} Chemical rate equations Let y(t) be t he concentration of a substance in a chemical reaction (typical units are moles/liter). The change in the concentration, under appropriate conditions, is modeled by the equation dy/dt=-ky^n for t≥0, where k>0 is a rate constant and the positive integer n is the order of the reaction.
b. Solve the initial value problem for a second-order reaction (n=2) assuming y(0)=y0.
17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.
The population increases from 50 to 60 in the first month and eventually levels off at 150.
20–22. {Use of Tech} Solving the Gompertz equation Solve the Gompertz equation in Exercise 19 with the given values of r, K, and M₀. Then graph the solution to be sure that M(0) and lim(t→∞) M(t) are correct.
r = 0.05, K = 1200, M₀ = 90
[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.
c. Graph the solutions in part (b) and describe their behavior as t increases.