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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

In Problems 27 through 31, a function y = g(x) is described by some geometric property of its graph. Write a differential equation of the form dy/dx = f (x, y) having the function g as its solution
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.y + xy' = 2e2x
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.(2x sin y cos y) y' = 4x2 + sin2 y
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.(2x + 1)y' + y = (2x + 1)3/2
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (2x + 3y) dx + (3x + 2y) dy = 0
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.(x + ey) y' = xe-y -1
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.y' = √x + y
In Problems 27 through 31, a function y = g(x) is described by some geometric property of its graph. Write a differential equation of the form dy/dx = f (x, y) having the function g as its solution
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (x³ + ²) dx + (y² + lnx) dy = 0
Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (2xy2 + 3x2) dx + (2x2y + 4y³) dy = 0
Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general
If c ≠ 0, verify that the function defined by y(x) x/(cx - 1) (with the graph illustrated in Fig. 1.3.26) sat- isfies the differential equation x2y' + y2 = 0 if x ≠ 1/c. Sketch a variety of such
Problems 33 and 34 explore vertical motion on a planet with gravitational acceleration different than the Earth’s.A person can throw a ball straight upward from the surface of the earth to a
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (3x² + 2y2) dx + (4xy + 6y2) dy = 0
In Problems 32 through 36, write—in the manner of Eqs. (3) through (6) of this section—a differential equation that is a mathematical model of the situation described.The acceleration dv=dt of a
In a certain culture of bacteria, the number of bacteria increased sixfold in 10 h. How long did it take for the population to double?
Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general
Problems 33 and 34 explore vertical motion on a planet with gravitational acceleration different than the Earth’s.On the planet Gzyx, a ball dropped from a height of 20 ft hits the ground in 2 s.
In Problems 32 through 36, write—in the manner of Eqs. (3) through (6) of this section—a differential equation that is a mathematical model of the situation described.The time rate of change of
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (4x - y) dx + (6y-x) dy = 0
Problems 29 through 32 explore the connections among general and singular solutions, existence, and uniqueness.Find a general solution and any singular solutions of the differential equation dy/dx =
Problems 30 through 32 explore the relation between the speed of an auto and the distance it skids when the brakes are applied.Suppose that a car skids 15 m if it is moving at 50 km/h when the brakes
In Problems 32 through 36, write—in the manner of Eqs. (3) through (6) of this section—a differential equation that is a mathematical model of the situation described.The time rate of change of a
Carry out an investigation similar to that in Problem 30, except with the differential equation y' = +√1- y2. Does it suffice simply to replace cos(x - c) with sin(x - c) in piecing together a
Problems 30 through 32 explore the relation between the speed of an auto and the distance it skids when the brakes are applied.The skid marks made by an automobile indicated that its brakes were
Discuss the difference between the differential equations (dy/dx)2 = 4y and dy/dx = 2√y. Do they have the same solution curves? Why or why not? Determine the points (a, b) in the plane for which
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. 2x5/2-3y5/3 2x5/2y2/3 - dx + 3y5/3-2x5/2 3x3/2y5/3 - dy = 0
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).yy" + (y')2 = 0 Example 11 Solve the
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (ex sin y + tan y) dx + (ex cos y + x sec² y) dy = 0
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).xy" = y' Example 11 Solve the
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. 2x y 3y² x4 dx + + -15 dy = 0
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (x + tan-¹ y) dx + x + y 1+ y2 dy = 0
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. 3,2 (3x²y³ + y4) dx + (3x³y² + y² + 4xy³) dy = 0
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (cosx + Iny) dx + G ( + e ²³ ) dy = 0
Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general
In Problems 31 through 42, verify that the given differential equation is exact; then solve it. (1+ yexy) dx + (2y + xexy) dy = 0
Suppose that sodium pentobarbital is used to anesthetize a dog. The dog is anesthetized when its bloodstream contains at least 45 milligrams (mg) of sodium pentobarbitol per kilogram of the dog’s
Suppose that you discover in your attic an overdue library book on which your grandfather owed a fine of 30 cents 100 years ago. If an overdue fine grows exponentially at a 5% annual rate compounded
Each of the differential equations in Problems 31 through 36 is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general
In Problems 32 through 36, write—in the manner of Eqs. (3) through (6) of this section—a differential equation that is a mathematical model of the situation described.In a city with a fixed
Problems 33 through 37 illustrate the application of linear first-order differential equations to mixture problems.A tank initially contains 60 gal of pure water. Brine containing 1 lb of salt per
Carbon taken from a purported relic of the time of Christ contained 4.6 x 1010 atoms of 14C per gram. Carbon extracted from a present-day specimen of the same substance contained 5.0 x 1010 atoms of
Problems 33 through 37 illustrate the application of linear first-order differential equations to mixture problems.Rework Example 4 for the case of Lake Ontario, which empties into the St. Lawrence
A stone is dropped from rest at an initial height h above the surface of the earth. Show that the speed with which it strikes the ground is v = √2gh.
Carbon extracted from an ancient skull contained only one sixth as much 14C as carbon extracted from present-day bone. How old is the skull?
In Problems 32 through 36, write—in the manner of Eqs. (3) through (6) of this section—a differential equation that is a mathematical model of the situation described.In a city having a fixed
Problems 43 through 46 concern the differential equationwhere k is a constant.Suppose the velocity v of a motorboat coasting in water satisfies the differential equation dv/dt = kv2. The initial
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).xy" + y' = 4x Example 11 Solve the
Problems 43 through 46 concern the differential equationwhere k is a constant.Suppose a population P of rodents satisfies the differential equation dP/dt = kP2. Initially, there are P(0) = 2 rodents,
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).y" = (y')2 Example 11 Solve the
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).x2y" + 3xy' = 2 Example 11 Solve the
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).yy" + (y')2 = yy' Example 11 Solve
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).y" = (x + y')2 Example 11 Solve the
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).y3y" = 1 Example 11 Solve the
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).y" = 2y (y')3 Example 11 Solve the
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).y" = 2yy' Example 11 Solve the
Find a general solution of each reducible second-order differential equation in Problems 43–54. Assume x, y and/or y' positive where helpful (as in Example 11).yy" = 3(y')2 Example 11 Solve the
Suppose that n ≠ 0 and n ≠ 1. Show that the substitution v = y1-n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation. dv dx +(1-n)P(x)v(x)=(1-n)Q(x).
Problems 54 through 64 illustrate the application of Torricelli’s law.A water tank has the shape obtained by revolving the curve y = x4/3 around the y-axis. A plug at the bottom is removed at 12
Show that the substitution v = ln y transforms the differential equation dy/dx + P(x)y = Q(x)(y ln y) into the linear equation dv/dx + P(x) = Q(x)v(x).
Problems 54 through 64 illustrate the application of Torricelli’s law.At time t = 0 the bottom plug (at the vertex) of a full conical water tank 16 ft high is removed. After 1 h the water in the
Show that the substitution v = ax + by + c transforms the differential equation dy/dx = F(ax + by + c) into a separable equation.
Use the idea in Problem 57 to solve the equationProblem 57Show that the substitution v = ln y transforms the differential equation dy/dx + P(x)y = Q(x)(y ln y) into the linear equation dv/dx + P(x) =
Use the method in Problem 59 to solve the differential equationProblem 59Solve the differential equationby finding h and k so that the substitutions x = u + h, y = v + k transform it into the
Problems 54 through 64 illustrate the application of Torricelli’s law.A cylindrical tank with length 5 ft and radius 3 ft is situated with its axis horizontal. If a circular bottom hole with a
Make an appropriate substitution to find a solution of the equation dy/dx = sin(x - y). Does this general solution contain the linear solution y(x) = x - π/2 that is readily verified by substitution
Show that the solution curves of the differential equationare of the form x3 + y3 = Cxy. dy dx =- || y (2x³ - y³) x(2y3x3)
Use the method of Problem 63 to solve the equations in Problems 64 and 65, given that y1(x) = x is a solution of each.Problem 63The equation dy/dx = A(x)y2 + B(x)y + C(x) is called a Riccati
An equation of the formis called a Clairaut equation. Show that the one-parameter family of straight lines described byis a general solution of Eq. (37). y = xy' + g(y) (37)
Consider the Clairaut equationfor which g(y') = -1/4 (y')2 in Eq. (37). Show that the lineis tangent to the parabolaExplain why this implies that y = x2 is a singular solution of the given Clairaut
Derive Eq. (18) in this section from Eqs. (16) and (17). In (v + √√1 + v²) = -k ln x + C, (16)
In the situation of Example 7, suppose that a = 100 mi, v0 = 400 mi/h, and w = 40 mi/h. Now how far northward does the wind blow the airplane? Example 7 Flight trajectory If a = 200 mi, vo = 500
As in the text discussion, suppose that an airplane maintains a heading toward an airport at the origin. If v0 = 500 mi/h and w = 50 mi/h (with the wind blowing due north), and the plane begins at
In the calculus of plane curves, one learns that the curvature k of the curve y = y(x) at the point (x, y) is given byand that the curvature of a circle of radius r is k = 1/r.Conversely, substitute
A river 100 ft wide is flowing north at w feet per second. A dog starts at (100, 0) and swims at v0 = 4 ft/s, always heading toward a tree at (0, 0) on the west bank directly across from the dog’s
In Problems 13 through 18, use a computer system or graphing calculator to plot a slope field and/or enough solution curves to indicate the stability or instability of each critical point of the
In Problems 13 through 18, use a computer system or graphing calculator to plot a slope field and/or enough solution curves to indicate the stability or instability of each critical point of the
In Problems 13 through 18, use a computer system or graphing calculator to plot a slope field and/or enough solution curves to indicate the stability or instability of each critical point of the
In Problems 13 through 18, use a computer system or graphing calculator to plot a slope field and/or enough solution curves to indicate the stability or instability of each critical point of the
Suppose that the population P(t) of a country satisfies the differential equation dP/dt = kP(200 - P) with k constant. Its population in 1960 was 100 million and was then growing at the rate of 1
Consider the differential equation dx/dt = kx - x3. (a) If k ≦ 0, show that the only critical value c = 0 of x is stable.(b) If k > 0, show that the critical point c = 0 is now unstable, but
Consider two population functions P1(t) and P2(t), both of which satisfy the logistic equation with the same limiting population M but with different values k1 and k2 of the constant k in Eq. (3).
Derive the solutionof the logistic initial value problem P' = kP(M - P), P(0) = P0. Make it clear how your derivation depends on whether 0 0 0 > M. P(t)= MPo Po+(M-Po)e-kMt
(a) Derive the solutionof the extinction-explosion initial value problem P' = kP(P - M), P(0) = P0.(b) How does the behavior of P(t) as t increases depend on whether 0 0 0 > M? P(t)= MPo Po
If P(t) satisfies the logistic equation in (3), use the chain rule to show thatConclude that P'' > 0 if 0 0 if P > M. In particular, it follows that any solution curve that crosses the line P
The data in the table in Fig. 2.1.7 are given for a certain population P(t) that satisfies the logistic equation in (3).(a) What is the limiting population M? (Suggestion: Use the approximationwith h
This problem deals with the differential equation dx/dt = kx(x - M) - h that models the harvesting of an unsophisticated population (such as alligators). Show that this equation can be rewritten in
During the period from 1790 to 1930, the U.S. population P(t) (t in years) grew from 3.9 million to 123.2 million. Throughout this period, P(t) remained close to the solution of the initial value
Use the alternative formsof the solution in (15) to establish the conclusions stated in (17) and (18). x (t) = N(xo-H) + H(N-xo)e-k(N-H)t (xo-H)+(N-xo)e-k(N-H)t H(N-xo)e-k(N-H)t - N(H - xo) (N −
Separate variables in the logistic harvesting equation dx/dt = k(N - x)(x - H) and then use partial fractions to derive the solution given in Eq. (15). x(t)
To solve the two equations in (10) for the values of k and M, begin by solving the first equation for the quantity x = e-50kM and the second equation for x2 = e-100kM. Upon equating the two resulting
A 30-year-old woman accepts an engineering position with a starting salary of $30,000 per year. Her salary S(t) increases exponentially, with S(t) = 30et/20 thousand dollars after t years. Meanwhile,
Suppose that in the cascade shown in Fig. 1.5.5, tank 1 initially contains 100 gal of pure ethanol and tank 2 initially contains 100 gal of pure water. Pure water flows into tank 1 at 10 gal/min, and
The half-life of radioactive cobalt is 5.27 years. Suppose that a nuclear accident has left the level of cobalt radiation in a certain region at 100 times the level acceptable for human habitation.
(a) Use the direction field of Problem 6 to estimate the values at x = 2 of the two solutions of the differential equation y' = x - y + 1 with initial values y(-3) = -0.2 and y(-3) = +0.2.(b) Use a
(a) Use the direction field of Problem 5 to estimate the values at x = 1 of the two solutions of the differential equation y' = y - x + 1 with initial values y(-1) = -1.2 and y(-1) = -0.8.(b) Use a
A certain city had a population of 25,000 in 1960 and a population of 30,000 in 1970. Assume that its population will continue to grow exponentially at a constant rate. What population can its city
Verify that if c > 0, then the function defined piecewise bysatisfies the differential equation y' = 4x √y for all x. Sketch a variety of such solution curves for different values of c. Then
In Problems 27 through 31, a function y = g(x) is described by some geometric property of its graph. Write a differential equation of the form dy/dx = f (x, y) having the function g as its solution

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