Use Euler’s method with n = 6 on the interval 0 ≤ t ≤ 3 to approximate the solution f (t) to y ’ = .1 y(20 - y), y(0) = 2.
> Solve the differential equation with the given initial condition. dN/dt = 2tN2, N(0) = 5
> Solve the differential equation with the given initial condition. dy/dx = ln x/√xy, y(1) = 4
> Solve the differential equation with the given initial condition. y ’ = t2/y , y(0) = -5
> Solve the differential equation with the given initial condition. y ’ = 5ty - 2t, y(0) = 1
> Solve the differential equation with the given initial condition. dy/dt = [(1 + t)/(1 + y)]2, y(0) = 2
> Solve the differential equation with the given initial condition. dy/dt = (t + 1)/ty, t > 0, y(1) = -3
> Solve the differential equation with the given initial condition. y' = -y2 sin t, y(π/2) = 1
> Solve the differential equation with the given initial condition. 3y2y’ = -sin t, y(π/2) = 1
> Solve the differential equation with the given initial condition. y ’ = t2 e-3y, y(0) = 2
> Solve the differential equation with the given initial condition. y2y’ = t cos t, y(0) = 2
> Let y = y(t) be the downward speed (in feet per second) of a skydiver after t seconds of free fall. This function satisfies the differential equation y ’ = .2(160 - y), y(0) = 0. What is the skydiver’s acceleration when her downward speed is 60 feet per
> Solve the differential equation with the given initial condition. y ’ = y2 - e3t y2, y(0) = 1
> Solve the differential equation with the given initial condition. y ’ = 2te-2y - e-2y, y(0) = 3
> Solve the following differential equations: yy’ = t sin(t2 + 1)
> Solve the following differential equations: y’ = (y - 3)2 ln t
> Solve the following differential equations: y2y’ = tan t
> Solve the following differential equations: y’ = ln t / ty
> Solve the following differential equations: y’ = 1/(ty + y)
> Solve the following differential equations: y ey = tet2
> Solve the following differential equations: (1 + t2) y’ = ty2
> Solve the following differential equations: y = 3t2 y2
> If the function f (t) is a solution of the initial-value problem y’ = et + y, y(0) = 0, find f (0) and f ‘ (0).
> Solve the following differential equations: y’ = (et/y)2
> Solve the following differential equations: y ’ = √(y/t)
> Solve the following differential equations: y’ = e4yt3 - e4y
> Solve the following differential equations: y ’ = (t/y)2 et3
> Solve the following differential equations: dy/dt = t2y2 / (t3 + 8)
> Solve the following differential equations: dy/dt = t1/2y2
> Solve the following differential equations: dy/dt = -1/t2y2
> Refer to the differential equation in Exercise 39. (a) Obviously, if you start with zero fish, f (t) = 0 for all t. Confirm this on the slope field. Are there any other constant solutions? (b) Describe the population of fish if the initial population is
> Let f (t) denote the number (in thousands) of fish in a lake after t years, and suppose that f (t) satisfies the differential equation y' = 0.1y (5 - y). The slope field for this equation is shown in Fig. 9. (a) With the help of the slope field, discus
> When a certain liquid substance A is heated in a flask, it decomposes into a substance B at such a rate (measured in units of A per hour) that at any time t is proportional to the square of the amount of substance A present. Let y = f (t) be the amount o
> If the function f (t) is a solution of the initial-value problem y ’ = 2y - 3, y(0) = 4, find f (0) and f ‘ (0).
> The Gompertz growth equation is dy/dt = -ay ln y/b, where a and b are positive constants. This equation is used in biology to describe the growth of certain populations. Find the general form of solutions to this differential equation. (Figure 8 shows
> Some homeowner’s insurance policies include automatic inflation coverage based on the U.S. Commerce Department’s construction cost index (CCI). Each year, the property insurance coverage is increased by an amount based on the change in the CCI. Let f (t)
> Mothballs tend to evaporate at a rate proportional to their surface area. If V is the volume of a mothball, then its surface area is roughly a constant times V2/3. So the mothball’s volume decreases at a rate proportional to V2/3. Suppose that initially
> In certain learning situations a maximum amount, M, of information can be learned, and at any time, the rate of learning is proportional to the amount yet to be learned. Let y = f (t) be the amount of information learned up to time t. Construct and solve
> Is the constant function f (t) = -4 a solution of the differential equation y’ = t2 (y + 4)?
> Is the constant function f (t) = 3 a solution of the differential equation y’ = 6 - 2y?
> State the order of the differential equation and verify that the given function is a solution. (1 - t2)y’’ - 2ty ’ + 6y = 0, y(t) = ½ (3t2 - 1)
> Let t represent the total number of hours that a truck driver spends during a year driving on a certain highway connecting two cities, and let p(t) represent the probability that the driver will have at least one accident during these t hours. Then, 0 ≤
> Show that the function f (t) = (e-t + 1)-1 satisfies y’ + y2 = y, y(0) = ½.
> Show that the function f (t) = 5 e2t satisfies y’’ - 3y’ + 2y = 0, y(0) = 5, y’ (0) = 10.
> Show that the function f (t) = t2 – ½ is a solution of the differential equation (y’ )2 - 4y = 2.
> Show that the function f (t) = 3/2 et2 – ½ is a solution of the differential equation y’ - 2ty = t.
> Solve the differential equation. y’ = y/t - 3y, t > 0
> Solve the differential equation. y ’ / (t + 1) = y + 1
> Solve the differential equation. y2y’ = 4t3 - 3t2 + 2
> Use Euler’s method with n = 5 on the interval 0 ≤ t ≤ 1 to approximate the solution f (t) to y ’ = ½ y (y - 10), y(0) = 9.
> Let f (t) be the solution to y ’ = (t + 1)/y, y(0) = 1. Use Euler’s method with n = 3 on 0 ≤ t ≤ 1 to estimate f (1). Then, show that Euler’s method gives the exact value of f (1) by solving the differential equation.
> The function f (t) = 5000/(1 + 49e-t) is the solution of the differential equation y’ = .0002y(5000 - y) from Example 8. (a) Graph the function in the window [0, 10] by [-750, 5750]. (b) In the home screen, compute .0002 f (3)(5000 - f
> Let f (t) be the solution to y ’ = 2e2t-y, y(0) = 0. Use Euler’s method with n = 4 on 0 ≤ t ≤ 2 to estimate f (2). Then show that Euler’s method gives the exact value of f (2) by solving the differential equation.
> A continuous annuity of $12,000 per year is to be funded by steady withdrawals from a savings account that earns 6% interest compounded continuously. (a) What is the smallest initial amount in the account that will fund such an annuity forever? (b) What
> A bank account has $20,000 earning 5% interest compounded continuously. A pensioner uses the account to pay himself an annuity, drawing continuously at a $2000 annual rate. How long will it take for the balance in the account to drop to zero?
> Suppose that in a chemical reaction, each gram of substance A combines with 3 grams of substance B to form 4 grams of substance C. The reaction begins with 10 grams of A, 15 grams of B, and 0 grams of C present. Let y = f (t) be the amount of C present a
> The birth rate in a certain city is 3.5% per year, and the death rate is 2% per year. Also, there is a net movement of population out of the city at a steady rate of 3000 people per year. Let N = f (t) be the city’s population at time t. (a) Write a diff
> Sketch the solution of the differential equation. Also indicate the constant solution. y’ = y3 - 6y2 + 9y, y(0) = - 1/4, y(0) = 1/4, y(0) = 4
> Sketch the solution of the differential equation. Also indicate the constant solution. y ’ = .4y2(1 - y), y(0) = -1, y(0) = .1, y(0) = 2
> Sketch the solution of the differential equation. Also indicate the constant solution. y’ = 3/(y + 3), y(0) = 2
> Sketch the solution of the differential equation. Also indicate the constant solution. y’ = 1/ (y2 + 1), y(0) = -1
> Sketch the solution of the differential equation. Also indicate the constant solution. y’ = 1 + cos y, y(0) = - 3/4
> Consider the differential equation y ’ = .2(10 - y) from Example 6. If the initial temperature of the steel rod is 510˚, the function f (t) = 10 + 500 e-0.2t is the solution of the differential equation. (a) Graph the function in the window [0, 30] by [-
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = ln y, y(0) = 2
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = y2 - 2y + 1, y(0) = -1
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = y2 + y, y(0) = - 13
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = 5 + 4y - y2, y(0) = 1
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = 2 cos y, y(0) = 0
> Solve the initial-value problem y ’ = ey2 (cos y)(1 - ey-1), y(0) = 1.
> If f (t) is a solution of y = (2 - y) e-y, is f (t) increasing or decreasing at some value of t where f (t) = 3?
> Let P(t) denote the price in dollars of a certain commodity at time t in days. Suppose that the rate of change of P is proportional to the difference D - S of the demand D and supply S at any time t. Suppose further that the demand and supply are related
> Find a curve in the xy-plane passing through the origin and whose slope at the point (x, y) is x + y.
> Solve the differential equation. y – 1/2(1 + t) y = 1 + t, t ≥ 0
> Answer the question in Exercise 29 by using the differential equation to determine the sign of f ‘(t). Exercise 29: If y0 > 1, is the solution y = f (t) of the initial-value problem y’ = 2y (1 - y), y (0) = y0, dec
> Solve the differential equation. yy ’ + t = 6t2, y(0) = 7
> Solve the differential equation. y’ = 5 - 8y, y(0) = 1
> Solve the differential equation. yy ’ + t = 6t2, y(0) = 7
> Solve the differential equation. y’ = tet+y, y(0) = 0
> Solve the differential equation. y' = 7y ’ + ty’ , y(0) = 3
> Solve the differential equation. (y’ )2 = t
> What is a constant solution to a differential equation?
> What is a solution curve?
> What does it mean for a function to be a solution to a differential equation?
> What is a differential equation?
> If y0 > 1, is the solution y = f (t) of the initial-value problem y’ = 2y (1 - y), y (0) = y0, decreasing for all t > 0? Answer this question based on the slope field shown in Fig. 8. Figure 8: Y 3 2 1 0 0 1 2 3 4 t
> Describe Euler’s method for approximating the solution of a differential equation.
> What is the logistic differential equation?
> Outline the procedure for sketching a solution of an autonomous differential equation.
> How do you recognize an autonomous differential equation from its slope field?
> What is an autonomous differential equation?
> What is an integrating factor and how does it help you solve a first-order linear differential equation?
> What is a first-order linear differential equation?
> Describe the separation-of-variables technique for obtaining the solution to a differential equation.
> What is the slope field?
> Suppose that f (t) satisfies the initial-value problem y = y2 + ty - 7, y(0) = 3. Is f (t) increasing or decreasing at t = 0?
> Figure 8 shows a portion of the solution curve of the differential equation y’ = 2y (1 - y) through the point (0, 2). On Fig. 8 or a copy of it, draw an approximation of the solution curve of the differential equation yâ€
> Find two constant solutions of y ’ = 4y (y - 7).
> Suppose that f (t) is a solution of y ’ = t2 - y2 and the graph of f (t) passes through the point (2, 3). Find the slope of the graph when t = 2.
> Suppose that f (t) is a solution of the differential equation y ’ = ty - 5 and the graph of f (t) passes through the point (2, 4). What is the slope of the graph at this point?
> The differential equation y’ = 2ty + et2, y(0) = 5, has solution y = (t + 5)et2. In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values calcula
> The differential equation y’ = et - 2y, y(0) = 1, has solution y = 1/3 (2e-2t + et ). In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values ca
