Solve the following differential equations: y’ = (y - 3)2 ln t
> Consider the initial-value problem y ’ = - y/(1 + t) + 10, y(0) = 50. (a) Is the solution increasing or decreasing when t = 0? (b) Find the solution and plot it for 0 ≤ t ≤ 4.
> A certain piece of news is being broadcast to a potential audience of 200,000 people. Let f (t) be the number of people who have heard the news after t hours. Suppose that y = f (t) satisfies y' = .07(200,000 - y), y (0) = 10. Describe this initial-va
> Solve the initial-value problem. ty’ + y = sin t, y(π/2) = 0, t > 0
> Solve the initial-value problem. y’ + 2y cos(2t) = 2 cos(2t), y(π/2) = 0
> Solve the initial-value problem. ty’ - y = -1, y(1) = 1, t > 0
> Solve the initial-value problem. y ’ + y = e2t, y(0) = -1
> Solve the initial-value problem. y’ = 2(10 - y), y(0) = 1
> Solve the initial-value problem. y ’ + y/(1 + t) = 20, y(0) = 10, t ≥ 0
> Solve the initial-value problem. ty’ + y = ln t, y(e) = 0, t > 0
> Solve the initial-value problem. y' + 2y = 1, y(0) = 1
> Solve the given equation using an integrating factor. Take t > 0. 1 / √(t + 1) y’ + y = 1
> Solve the given equation using an integrating factor. Take t > 0. y’ + y = 2 - et
> Let f (t) be the balance in a savings account at the end of t years. Suppose that y = f (t) satisfies the differential equation y’ = .04y + 2000. (a) If after 1 year the balance is $10,000, is it increasing or decreasing at that time? At what rate is it
> Solve the given equation using an integrating factor. Take t > 0. et y’ + y = 1
> Solve the given equation using an integrating factor. Take t > 0. 6y’ + ty = t
> Solve the given equation using an integrating factor. Take t > 0. y ’ = e-t (y + 1)
> Solve the given equation using an integrating factor. Take t > 0. (1 + t)y ’ + y = -1
> Solve the given equation using an integrating factor. Take t > 0. y’ - 2y = e2t
> Solve the given equation using an integrating factor. Take t > 0. y’ + y/(10 + t) = 0
> Solve the given equation using an integrating factor. Take t > 0. y’ + y = e-t + 1
> Solve the given equation using an integrating factor. Take t > 0. y’ = .5(35 - y)
> Solve the given equation using an integrating factor. Take t > 0. y ’ = 2(20 - y)
> Solve the given equation using an integrating factor. Take t > 0. y’ - 2ty = -4t
> Let f (t) be the balance in a savings account at the end of t years, and suppose that y = f (t) satisfies the differential equation y ’ = .05y - 10,000. (a) If after 1 year the balance is $150,000, is it increasing or decreasing at that time? At what rat
> Solve the given equation using an integrating factor. Take t > 0. y’ + 2ty = 0
> Solve the given equation using an integrating factor. Take t > 0. y' + y = 1
> Find an integrating factor for each equation. Take t > 0. y’ = t2(y + 1)
> Find an integrating factor for each equation. Take t > 0. y ’ – y/(10 + t) = 2
> Find an integrating factor for each equation. Take t > 0. y ’ + √t y = 2(t + 1)
> Solve the following differential equations: dy/dt = et/ey
> Solve the following differential equations: dy/dt = te2y
> Solve the following differential equations: dy/dt = (5 – t)/y2
> One problem in psychology is to determine the relation between some physical stimulus and the corresponding sensation or reaction produced in a subject. Suppose that, measured in appropriate units, the strength of a stimulus is s and the intensity of the
> A model that describes the relationship between the price and the weekly sales of a product might have a form such as dy/dp = -1/2 (y/p + 3), where y is the volume of sales and p is the price per unit. That is, at any time, the rate of decrease of sale
> A lake is stocked with 100 fish. Let f (t) be the number of fish after t months, and suppose that y = f (t) satisfies the differential equation y’ = .0004y(1000 - y). Figure 7 shows the graph of the solution to this differential equatio
> 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: 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.
> 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.
> 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
