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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.x3y' = x2y - y3
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.4xy2 + y' = 5x4y2
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.y' + 3y = 3x2e-3x
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.x(x + y)y' + y(3x + y) = 0
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.y' = x2 - 2xy + y2
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.ex + yexy + (ey + xeyx) y' = 0
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.y' = (4x + y)2
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.2x2y - x3y' = y3
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.(x + y)y' = 1
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.x2y' + 2xy = 5y3
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" + 4y = 0 Example 11 Solve the
Consider the initially full hemispherical water tank of Example 8, except that the radius r of its circular bottom hole is now unknown. At 1 P.M. the bottom hole is opened and at 1:30 P.M. the depth
Separate variables and use partial fractions to solve the initial value problems in Problems 1–8. Use either the exact solution or a computer-generated slope field to sketch the graphs of several
Assume that a body moving with velocity v encounters resistance of the form dv/dt = -kv3/2. Show thatConclude that under a 3/2 -power resistance a body coasts only a finite distance before coming to
Suppose that a body moves through a resisting medium with resistance proportional to its velocity v, so that dv/dt = -kv.(a) Show that its velocity and position at time t are given by(b) Conclude
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.3x5y2 + x3y' = 2y2
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.xy' + 3y = 3x-3/2
A motorboat weighs 32,000 lb and its motor provides a thrust of 5000 lb. Assume that the water resistance is 100 pounds for each foot per second of the speed v of the boat. ThenIf the boat starts
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
It is proposed to dispose of nuclear wastes—in drums with weight W = 640 lb and volume 8 ft3—by dropping them into the ocean (v0 = 0). The force equation for a drum falling through water iswhere
Integrate the velocity function in Eq. (13) to obtain the upward-motion position function given in Eq. (14) with initial condition y(0) = y0. v (t) = = √tan (C₁-1 √/PB) with C₁ = tan¹ (UO
Suppose that a car starts from rest, its engine providing an acceleration of 10 ft/s2, while air resistance provides 0.1 ft/s2 of deceleration for each foot per second of the car’s velocity.(a)
Separate variables in Eq. (15) and substitute u = v √ρ/g to obtain the downward-motion velocity function given in Eq. (16) with initial condition v(0) = v0. * =-x + 3 =-x(1-4+). dv dt pv² (15)
If a ball is projected upward from the ground with initial velocity v0 and resistance proportional to v2, deduce from Eq. (14) that the maximum height it attains is ymax =l 20 In]1 + ρυτ to g
Separate variables in Eq. (12) and substitute u = v√ρ/g to obtain the upward-motion velocity function given in Eq. (13) with initial condition v(0) = v0. ap dt -g-pu² = -8 (1 + 20²). (12)
A woman bails out of an airplane at an altitude of 10,000 ft, falls freely for 20 s, then opens her parachute. How long will it take her to reach the ground? Assume linear air resistance ρv ft/s2,
Suppose that ρ = 0.075 (in fps units, with g = 32 ft/s2) in Eq. (15) for a paratrooper falling with parachute open. If he jumps from an altitude of 10,000 ft and opens his parachute
Integrate the velocity function in Eq. (16) to obtain the downward-motion position function given in Eq. (17) with initial condition y(0) = y0. √tan v(t) = tanh(Cz−t Vpg) with 1-¹ ( 10
According to a newspaper account, a paratrooper survived a training jump from 1200 ft when his parachute failed to open but provided some resistance by flapping unopened in the wind. Allegedly he hit
Suppose that the fish population P(t) in a lake is attacked by a disease at time t = 0, with the result that the fish cease to reproduce (so that the birth rate is β = 0) and the death rate ı
Suppose that the paratrooper of Problem 22 falls freely for 30 s with ρ = 0.00075 before opening his parachute. How long will it now take him to reach the ground?Problem 22Suppose that ρ =
(a) Suppose a projectile is launched vertically from the surface r = R of the earth with initial velocity v0 = √2GM/R, so v20 = k2/R where k2 = 2GM. Solve the differential equation dr/dt = k/√r
A motorboat starts from rest (initial velocity v(0) = v0 = 0). Its motor provides a constant acceleration of 4 ft/s2 but water resistance causes a deceleration of v2/400 ft/s2. Find v when t = 10 s,
An arrow is shot straight upward from the ground with an initial velocity of 160 ft/s. It experiences both the deceleration of gravity and deceleration v2/800 due to air resistance. How high in the
(a) Suppose that a body is dropped (v0 = 0) from a distance r0 > R from the earth's center, so its acceleration is dv/dt = -GM/r2. Ignoring air resistance, show that it reaches the height r <
Suppose that at time t = 0, half of a “logistic” population of 100,000 persons have heard a certain rumor, and that the number of those who have heard it is then increasing at the rate of 1000
Suppose that you are stranded—your rocket engine has failed—on an asteroid of diameter 3 miles, with density equal to that of the earth with radius 3960 miles. If you have enough spring in your
The mass of the sun is 329,320 times that of the earth and its radius is 109 times the radius of the earth.(a) To what radius (in meters) would the earth have to be compressed in order for it to
Suppose that a projectile is fired straight upward from the surface of the earth with initial velocity v0 < √2GM/R. Then its height y(t) above the surface satisfies the initial value
In Jules Verne’s original problem, the projectile launched from the surface of the earth is attracted by both the earth and the moon, so its distance r(t) from the center of the earth satisfies the
In Problems 21 and 22, first use the method of Example 2 to construct a slope field for the given differential equation. Then sketch the solution curve corresponding to the given initial condition.
In Problems 19 through 22, a particle starts at the origin and travels along the x-axis with the velocity function v(t) whose graph is shown in Figs. 1.2.6 through 1.2.9. Sketch the graph of the
The equation dy/dx = A(x)y2 + B(x)y + C(x) is called a Riccati equation. Suppose that one particular solution y1(x) of this equation is known. Show that the substitutiontransforms the Riccati
In Problems 1 through 10, an initial value problem and its exact solution y(x) are given. Apply Euler’s method twice to approximate to this solution on the interval [0, 1/2], first with step size h
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
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
Separate variables and use partial fractions to solve the initial value problems in Problems 1–8. Use either the exact solution or a computer-generated slope field to sketch the graphs of several
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.(x2 - 1)y' + (x - 1)y = 1
In Problems 1 through 10, an initial value problem and its exact solution y(x) are given. Apply Euler’s method twice to approximate to this solution on the interval [0, 1/2], first with step size h
In Problems 1 through 10, an initial value problem and its exact solution y(x) are given. Apply Euler’s method twice to approximate to this solution on the interval [0, 1/2], first with step size h
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.y' = y + y3
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.x2y' + 2xy = 5y4
Separate variables and use partial fractions to solve the initial value problems in Problems 1–8. Use either the exact solution or a computer-generated slope field to sketch the graphs of several
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.xy' = 6y + 12x4y2/3
In Problems 1 through 10, an initial value problem and its exact solution y(x) are given. Apply Euler’s method twice to approximate to this solution on the interval [0, 1/2], first with step size h
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
Separate variables and use partial fractions to solve the initial value problems in Problems 1–8. Use either the exact solution or a computer-generated slope field to sketch the graphs of several
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
Separate variables and use partial fractions to solve the initial value problems in Problems 1–8. Use either the exact solution or a computer-generated slope field to sketch the graphs of several
In Problems 1 through 10, an initial value problem and its exact solution y(x) are given. Apply Euler’s method twice to approximate to this solution on the interval [0, 1/2], first with step size h
Separate variables and use partial fractions to solve the initial value problems in Problems 1–8. Use either the exact solution or a computer-generated slope field to sketch the graphs of several
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
In Problems 1 through 10, an initial value problem and its exact solution y(x) are given. Apply Euler’s method twice to approximate to this solution on the interval [0, 1/2], first with step size h
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
In Problems 1 through 10, an initial value problem and its exact solution y(x) are given. Apply Euler’s method twice to approximate to this solution on the interval [0, 1/2], first with step size h
In Problems 1 through 10, an initial value problem and its exact solution y(x) are given. Apply Euler’s method twice to approximate to this solution on the interval [0, 1/2], first with step size h
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
In Problems 1 through 12 first solve the equation f (x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f (x) Then analyze the sign of f (x) to determine whether
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
Example 4 dealt with the case 4h > kM2 in the equation dx/dt = kx (M - x) - h that describes constant-rate harvesting of a logistic population. Problems 26 and 27 deal with the other cases.If 4h =
The differential equation dx/dt = 1/10x(10 -x) - h models a logistic population with harvesting at rate h. Determine (as in Example 6) the dependence of the number of critical points on the parameter
Example 4 dealt with the case 4h > kM2 in the equation dx/dt = kx (M - x) - h that describes constant-rate harvesting of a logistic population. Problems 26 and 27 deal with the other cases.If 4h
The time rate of change of a rabbit population P is proportional to the square root of P. At time t = 0 (months) the population numbers 100 rabbits and is increasing at the rate of 20 rabbits per
The time rate of change of an alligator population P in a swamp is proportional to the square of P. The swamp contained a dozen alligators in 1988, two dozen in 1998. When will there be four dozen
Use the method of Problem 36 to fit the logistic equation to the actual U.S. population data (Fig. 2.1.4) for the years 1850, 1900, and 1950. Solve the resulting logistic equation and compare the
Consider the two differential equationseach having the critical points a, b, and c; suppose that a < b < c. For one of these equations, only the critical point b is stable; for the other
The differential equation dx/dt = 1/100x(x - 5) + s models a population with stocking at rate s. Determine the dependence of the number of critical points c on the parameter s, and then construct the
Consider the differential equation dx/dt = x + kx3 containing the parameter k. Analyze (as in Problem 21) the dependence of the number and nature of the critical points on the value of k, and
Suppose that a community contains 15,000 people who are susceptible to Michaud’s syndrome, a contagious disease. At time t = 0 the number N(t) of people who have developed Michaud’s syndrome is
Fit the logistic equation to the actual U.S. population data (Fig. 2.1.4) for the years 1900, 1930, and 1960. Solve the resulting logistic equation, then compare the predicted and actual populations
Suppose that the number x(t) (with t in months) of alligators in a swamp satisfies the differential equation dx/dt = 0.0001x2 - 0.01x.(a) If initially there are 25 alligators in the swamp, solve this
Problems 23 through 28 explore the motion of projectiles under constant acceleration or deceleration.What is the maximum height attained by the arrow of part (b) of Example 3? (-a,
You bail out of the helicopter of Example 3 and pull the ripcord of your parachute. Now k = 1.6 in Eq. (3), so your downward velocity satisfies the initial value problemIn order to investigate your
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.xy' + 6y = 3xy4/3
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.ey + y cos x + (xey + sinx)y' = 0
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.2xy' + y3 e-2x = 2xy
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.9x2y2 + x3/2y' = y2
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.y2(xy' + y)(1 + x4)1/2 = x
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.2y + (x + 1)y' = 3x + 3
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.3y2y' + y3 = e-x
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.9x1/2 y4/3 - 12x1/5 y3/2 + (8x3/2 y1/3 - 15x6/5 y1/2) y'= 0
Solve the differential equation (dy/dx)2 = 4y to verify the general solution curves and singular solution curve that are illustrated in Fig. 1.4.5. Then determine the points (a, b) in the plane for
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.3xy2y' = 3x4 + y3
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
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x.3y + x3y4 + 3xy' = 0
Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout.xey y' = 2(ey + x3 e2x)

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