43–44. Motion in a gravitational field: An object is fired vertically upward with initial velocity v(0)=v₀ from initial position s(0)=s₀.

a. For the following values of v₀ and s₀, find the position and velocity functions for all times at which the object is above the ground (s = 0).

v₀ = 49 m/s, s₀ = 60 m

45–46. Harvesting problems Consider the harvesting problem in Example 6.

If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The general solution of the differential equation y'(t) = 1 is y(t) = t

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

y(t) = C₁ sin4t + C₂ cos4t; y''(t) + 16y(t) = 0

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁eᵗ + C₂teᵗ; u''(t) - 2u'(t) + u(t) = 0

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁t⁵ + C₂t⁻⁴ - t³; t²u''(t) - 20u(t) = 14t³

17–20. Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it.

y(t) = 8t⁶ - 3; ty'(t) - 6y(t) = 18, y(1) = 5

33–42. Solving initial value problems Solve the following initial value problems.​

y'(t) = 1 + eᵗ, y(0) = 4

Does the function y(t) = 2t satisfy the differential equation y'''(t) + y'(t) = 2?

Consider the differential equation y'(t)+9y(t)=10.

a. How many arbitrary constants appear in the general solution of the differential equation?

33–42. Solving initial value problems Solve the following initial value problems.​

y''(t) = teᵗ, y(0) = 0, y'(0) = 1

33–42. Solving initial value problems Solve the following initial value problems.​

y'(x) = 4 sec² 2x, y(0) = 8

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

y'(t) = t lnt + 1

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

y'(t) = 3 + e⁻²ᵗ

Another second-order equation Consider the differential equation y''(t) + k²y(t) = 0, where k is a positive real number.

c. Give the general solution of the equation for arbitrary k > 0 and verify your conjecture.

Explain why the graph of the solution to the initial value problem y'(t) = t²/(1 - t), y(-1) = ln 2 cannot cross the line t = 1.

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

u''(x) = 55x⁹ + 36x⁷ - 21x⁵ + 10x⁻³

A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.

a. Verify by substitution that when k = 1, a solution of the equation is y(t) = C₁eᵗ + C₂e⁻ᵗ. You may assume this function is the general solution.

A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.

d. For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C₁cosh(kt) + C₂sinh(kt), where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively (Section 7.3).

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

a. Verify by substitution that the general solution of the equation is P(t) = K/(1 + Ce⁻ʳᵗ), where C is an arbitrary constant.

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m'(t) + km(t) = I, where m(t) is the mass of the drug in the blood at time t ≥ 0, k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate.

b. Graph the solution for I = 10 mg/hr and k = 0.05 hr⁻¹.

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

​​{Use of Tech} Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v'(t) = g - bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.

c. Using the graph in part (b), estimate the terminal velocity lim(t→∞) v(t).

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

(Use of Tech) Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -kyⁿ(t), where y(t) is the concentration of the compound, for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.

c. Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions. 

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is:

dM/dt = -rM(t)ln(M(t)/K), M(0) = M₀,

where r and K are positive constants and 0 < M₀ < K.

b. Graph the solution for M₀ = 100 and r = 0.05.

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).

y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

c. Sketch the solution curve that corresponds to the initial condition y0=1. 

y′(t) = 2y + 4

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

c. Sketch the solution curve that corresponds to the initial condition y0=1. 

y′(t) = 6 - 2y

25–28. Two steps of Euler’s method For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.

y′(t) = 2−y, y(0) = 1; Δt = 0.1

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.

y′(t) = −y, y(0) = 1; y(t) = e⁻ᵗ

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.

y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²