1 Million+ Step-by-step solutions

Suppose that the tank in Example 7 is hemispherical, of radius R, initially full of water, and has an outlet of 5 cm^{2} cross sectional area at the bottom. (Make a sketch.) Set up the model for outflow. Indicate what portion of your work in Example 7 you can use (so that it can become part of the general method independent of the shape of the tank). Find the time t to empty the tank (a) for any R, (b) for R = 1 m. Plot t as function of R. Find the time when h = R/2 (a) for any R, (b) for R = 1 m.

A family of curves can often be characterized as the general solution of y' = f(x, y).

(a) Show that for the circles with center at the origin we get y' = -x/y.

(b) Graph some of the hyperbolas xy = c. Find an ODE for them.

(c) Find an ODE for the straight lines through the origin.

(d) You will see that the product of the right sides of the ODEs in (a) and (c) equals -1. Do you recognize this as the condition for the two families to be orthogonal (i.e., to intersect at right angles)? Do your graphs confirm this?

(e) Sketch families of curves of your own choice and find their ODEs. Can every family of curves be given by an ODE?

Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an initial condition is given, find the corresponding particular solution.

x^{3}dx + y^{3}dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an initial condition is given, find the corresponding particular solution.

e^{3θ}(dr + 3r dθ) = 0

Graph a direction field (by a CAS or by hand). In the field graph several solution curves by hand, particularly those passing through the given points (x, y).

y' = -2xy, (0, 1/2), (0, 1), (0, 2)

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.)

y' + y tan x = e^{-0.01x} cos x, y(0) = 0

Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.

y = √x + c

Solve the ODE by integration or by remembering a differentiation formula.

y''' = e^{-0.2x}

Show that for a linear ODE y' + p(x)y = r(x) with continuous p and r in |x - x_{0}| __<__ a Lipschitz condition holds. This is remarkable because it means that for a linear ODE the continuity of f(x, y) guarantees not only the existence but also the uniqueness of the solution of an initial value problem.

Find a general solution. Show the steps of derivation. Check your answer by substitution.

y' = (y + 4x)^{2} (Set y + 4x = v)

Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an initial condition is given, find the corresponding particular solution.

e^{x}(cos y dx - sin y dy) = 0

Graph a direction field (by a CAS or by hand). In the field graph several solution curves by hand, particularly those passing through the given points (x, y).

y' = sin^{2} y, (0, -0.4), (0, 1)

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.)

y' + 2y = 4 cos 2x, y(1/4π) = 3

Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.

xy = c

Solve the ODE by integration or by remembering a differentiation formula.

y'' = -y

Find a general solution. Show the steps of derivation. Check your answer by substitution.

y' = e^{2x-1 }y^{2}

(a) Apply the iteration to y' = x + y, y(0) = 0. Also solve the problem exactly.

(b) Apply the iteration to y' = 2y^{2}, y(0) = 1. Also solve the problem exactly.

(c) Find all solutions of y' = 2√y, y(1) = 0. Which of them does Picard’s iteration approximate?

(d) Experiment with the conjecture that Picard’s iteration converges to the solution of the problem for any initial choice of y in the integrand in (7) (leaving y_{0} outside the integral as it is). Begin with a simple ODE and see what happens. When you are reasonably sure, take a slightly more complicated ODE and give it a try.

3(y + 1) dx = 2x dy, (y + 1)x^{-4}

Graph a direction field (by a CAS or by hand). In the field graph several solution curves by hand, particularly those passing through the given points (x, y).

y' = 2y - y^{2}, (0, 0), (0, 1), (0, 2), (0, 3)

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.)

y' = 2y - 4x

Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.

y = x^{2} + c

Solve the ODE by integration or by remembering a differentiation formula.

y' = -1.5y

Find a general solution. Show the steps of derivation. Check your answer by substitution.

y' sin 2πx = πy cos 2πx

What happens in Prob. 2 if you replace y(2) = 1 with y(2) = k?

**Data from prob 2.**

Does the initial value problem (x - 2)y' = y, y(2) = 1 have a solution? Does your result contradict our present theorems?

Graph a direction field (by a CAS or by hand). In the field graph several solution curves by hand, particularly those passing through the given points (x, y).

yy' + 4x = 0, (1, 1), (2, 1/2)

Represent the given family of curves in the form G(x, y; c) = 0 and sketch some of the curves.

All circles with centers on the cubic parabola y = x^{3 }and passing through the origin (0, 0)

Find a general solution. Show the steps of derivation. Check your answer by substitution.

y^{3}y' + x^{3 }= 0

Solve the ODE by integration or by remembering a differentiation formula.

y' + xe^{-x}^{2}^{/2} = 0

y dx + [y + tan (x + y)] dy = 0, cos (x + y)

Find a general solution. Show the steps of derivation. Check your answer by substitution.

xy' = x + y (Set y/x = u)

Find all initial conditions such that (x^{2} - x)y' = (2x - 1)y has no solution, precisely one solution, and more than one solution.

(a) Verify that y is a solution of the ODE.

(b) Determine from y the particular solution of the IVP

(c) Graph the solution of the IVP.

y' + 5xy = 0, y = ce^{-2.5x}^{2}, y(0) = π

Verify that y is a solution of the ODE. (b) Determine from y the particular solution of the IVP. (c) Graph the solution of the IVP.

y' tan x = 2y - 8, y = c sin^{2} x + 4, y(1/2π) = 0

y' cos x + (3y - 1) sec x = 0, y(1/4π) = 4/3

Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.

x^{2} + (y - c)^{2} = c^{2}

Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare.

y' = 1 - y^{2}

(2xy dx + dy)e^{x2} = 0, y(0) = 2

Solve the IVP. Show the steps of derivation, beginning with the general solution.

y' = 1 + 4y^{2}, y(1) = 0

(a) Verify that y is a solution of the ODE.

(b) Determine from y the particular solution of the IVP

(c) Graph the solution of the IVP.

yy' = 4x, y^{2} - 4x^{2 }= c (y > 0), y(1) = 4

xy' + 4y = 8x^{4}, y(1) = 2

Model the motion of a body B on a straight line with velocity as given, y(t) being the distance of B from a point y = 0 at time t. Graph a direction field of the model (the ODE). In the field sketch the solution curve satisfying the given initial condition. Product of velocity times distance constant, equal to 2, y(0) = 2.

Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare.

xy' = y + x^{2}

(a + 1)y dx + (b + 1)x dy = 0, y(1) = 1, F = x^{a}y^{b}

Solve the IVP. Show the steps of derivation, beginning with the general solution.

dr/dt = -2tr, r(0) = r_{0}

Find the conditions under which the orthogonal trajectories of families of ellipses x^{2}/a^{2} + y^{2}/b^{2} = c are again conic sections. Illustrate your result graphically by sketches or by using your CAS. What happens if a → 0? If b → 0?

(a) Solve the ODE y' - y/x = -x^{-1 }cos (1/x). Find an initial condition for which the arbitrary constant becomes zero. Graph the resulting particular solution, experimenting to obtain a good figure near x = 0.

(b) Generalizing (a) from n = 1 to arbitrary n, solve the ODE y' - ny/x = -x^{n-2} cos (1/x). Find an initial condition as in (a) and experiment with the graph.

Model the motion of a body B on a straight line with velocity as given, y(t) being the distance of B from a point y = 0 at time t. Graph a direction field of the model (the ODE). In the field sketch the solution curve satisfying the given initial condition.

Square of the distance plus square of the velocity equal to 1, initial distance 1/√2

Discuss direction fields as follows.

(a) Graph portions of the direction field of the ODE (2) (see Fig. 7), for instance, -5__<__x__<__2, -1__<__y__<__5. Explain what you have gained by this enlargement of the portion of the field.

(b) Using implicit differentiation, find an ODE with the general solution x^{2}+ 9y^{2}= c (y > 0).^{ }Graph its direction field. Does the field give the impression that the solution curves may be semi-ellipses? Can you do similar work for circles? Hyperbolas? Parabolas? Other curves?

(c) Make a conjecture about the solutions of y' = -x/y from the direction field.

(d) Graph the direction field of y' = -1/2y and some solutions of your choice. How do they behave? Why do they decrease for y > 0?

(a) Graph portions of the direction field of the ODE (2) (see Fig. 7), for instance, -5

(b) Using implicit differentiation, find an ODE with the general solution x

(c) Make a conjecture about the solutions of y' = -x/y from the direction field.

(d) Graph the direction field of y' = -1/2y and some solutions of your choice. How do they behave? Why do they decrease for y > 0?

Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare. In Prob. 16 use Euler’s method.

Solve y' = y - y^{2}, y(0) = 0.2 by Euler’s method (10 steps, h = 0.1). Solve exactly and compute the error.

Solve the IVP. Show the steps of derivation, beginning with the general solution.

y' = (x + y - 2)^{2}, y(0) = 2, (Set v = x + y - 2)

An ODE may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution. The ODE y'^{2}- xy' + y = 0 is of this kind. Show by differentiation and substitution that it has the general solution y = cx - c^{2 }and the singular solution y = x^{2}/4. Explain Fig. 6.

If y' = f(x) with f independent of y, show that the curves of the corresponding family are congruent, and so are their OTs.

These properties are of practical and theoretical importance because they enable us to obtain new solutions from given ones. Thus in modeling, whenever possible, we prefer linear ODEs over nonlinear ones, which have no similar properties.

Show that nonhomogeneous linear ODEs (1) and homogeneous linear ODEs (2) have the following properties. Illustrate each property by a calculation for two or three equations of your choice. Give proofs.

y = 0 (that is, y(x) = 0 for all x, also written y(x) = 0) is a solution of (2) [not of (1) if r(x) ≠ 0!], called the trivial solution.

This is the simplest method to explain numerically solving an ODE, more precisely, an initial value problem (IVP). Using the method, to get a feel for numerics as well as for the nature of IVPs, solve the IVP numerically with a PC or a calculator, 10 steps. Graph the computed values and the solution curve on the same coordinate axes.

y' = y, y(0) = 1, h = 0.01

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.

y' - 0.4y = 29 sin x

Graph particular solutions of the following ODE, proceeding as explained.

(21) dy - y^{2}sin x dx = 0

(21) dy - y

(a) Show that (21) is not exact. Find an integrating factor using either Theorem 1 or 2. Solve (21).

(b) Solve (21) by separating variables. Is this simpler than (a)?

(c) Graph the seven particular solutions satisfying the following initial conditions y(0) = 1, y(Ï€/2) = Â± 1/2, Â± 2/3, Â± 1 (see figure below).

(d) Which solution of (21) do we not get in (a) or (b)?

Introduce limits of integration in (3) such that y obtained from (3) satisfies the initial condition y(x_{0}) = y_{0}.

Radium ^{224}_{88} Ra has a half-life of about 3.6 days.

(a) Given 1 gram, how much will still be present after 1 day?

(b) After 1 year?

These properties are of practical and theoretical importance because they enable us to obtain new solutions from given ones. Thus in modeling, whenever possible, we prefer linear ODEs over nonlinear ones, which have no similar properties.

Show that nonhomogeneous linear ODEs (1) and homogeneous linear ODEs (2) have the following properties. Illustrate each property by a calculation for two or three equations of your choice. Give proofs.

The difference of two solutions of (1) is a solution of (2).

This is the simplest method to explain numerically solving an ODE, more precisely, an initial value problem (IVP). Using the method, to get a feel for numerics as well as for the nature of IVPs, solve the IVP numerically with a PC or a calculator, 10 steps. Graph the computed values and the solution curve on the same coordinate axes.

y' = -5x^{4}y^{2}, y(0) = 1, h = 0.2, Sol. y = 1/(1 + x)^{5}

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.

y' = ay + by^{2} (a ≠ 0)

(a) If the birth rate and death rate of the number of bacteria are proportional to the number of bacteria present, what is the population as a function of time.

(b) What is the limiting situation for increasing time? Interpret it.

The efficiency of the engines of subsonic airplanes depends on air pressure and is usually maximum near 35,000 ft. Find the air pressure y(x) at this height. Physical information. The rate of change y'(x) is proportional to the pressure. At 18,000 ft it is half its value y_{0} = y(0) at sea level. Remember from calculus that if y = e^{kx}, then y' = ke^{kx} = ky. Can you see without calculation that the answer should be close to y_{0}/4?

These properties are of practical and theoretical importance because they enable us to obtain new solutions from given ones. Thus in modeling, whenever possible, we prefer linear ODEs over nonlinear ones, which have no similar properties.

Show that nonhomogeneous linear ODEs (1) and homogeneous linear ODEs (2) have the following properties. Illustrate each property by a calculation for two or three equations of your choice. Give proofs.

If y_{1} and y_{2} are solutions of y'_{1} + py_{1} = r_{1} and y'_{2 }+ py_{2} = r_{2}_{, }respectively (with the same p!), what can you say about the sum y_{1} + y_{2}?

Solve the IVP. Indicate the method used. Show the details of your work.

y' + 4xy = e^{-2x}^{2}, y(0) = -4.3

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it.

y' + y = y^{2}, y(0) = -1/3

Solve the IVP. Indicate the method used. Show the details of your work.

y' + 1/2y = y^{3}, y(0) = 1/3

A tank contains 400 gal of brine in which 100 lb of salt are dissolved. Fresh water runs into the tank at a rate of 2 gal/min. The mixture, kept practically uniform by stirring, runs out at the same rate. How much salt will there be in the tank at the end of 1 hour?

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it.

y' + y = -x/y

Solve the IVP. Indicate the method used. Show the details of your work.

x sinh y dy = cosh y dx, y(3) = 0

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it.

y' = (tan y)/(x - 1), y(0) = 1/2π

The tank in Fig. 28 contains 80 lb of salt dissolved in 500 gal of water. The inflow per minute is 20 lb of salt dissolved in 20 gal of water. The outflow is 20 gal/min of the uniform mixture. Find the time when the salt content y(t) in the tank reaches 95% of its limiting value (as t â†’ âˆž).

Could you see, practically without calculation, that the answer in Prob. 27 must lie between 60 and 70 min? Explain.

**Data from Prob 27**

If a wet sheet in a dryer loses its moisture at a rate proportional to its moisture content, and if it loses half of its moisture during the first 10 min of drying, when will it be practically dry, say, when will it have lost 99% of its moisture? First guess, then calculate.

2xyy' + (x - 1)y^{2 }= x^{2}e^{x}, (Set y^{2 }= z)

A metal bar whose temperature is is 20°C placed in boiling water. How long does it take to heat the bar to practically 100°C say, to 99.9°C, if the temperature of the bar after 1 min of heating is 51.5°C? First guess, then calculate.

A rocket is shot straight up from the earth, with a net acceleration (= acceleration by the rocket engine minus gravitational pullback) of 7t m/sec^{2 }during the initial stage of flight until the engine cut out at t = 10 sec. How high will it go, air resistance neglected?

A Riccati equation is of the form

y' + p(x)y = g(x)y^{2} + h(x)

A Clairaut equation is of the form

y = xy' + g(y').

(a) Apply the transformation y = Y + 1/u to the Riccati equation (14), where Y is a solution of (14), and obtain for u the linear ODE u' + (2Yg - p)u = -g. Explain the effect of the transformation by writing it as y = Y + v, v = 1/u.

(b) Show that y = Y = x is a solution of the ODE y' - (2x^{3} + 1) y = -x^{2}y^{2} - x^{4} - x + 1 and solve this Riccati equation, showing the details.

(c) Solve the Clairaut equation y^{'2} - xy' + y = 0 as follows. Differentiate it with respect to x, obtaining y''(2y' - x) = 0. Then solve (A) y'' = 0 and (B) 2y' - x = 0 separately and substitute the two solutions (a) and (b) of (A) and (B) into the given ODE. Thus obtain (a) a general solution (straight lines) and (b) a parabola for which those lines (a) are tangents (Fig. 6 in Prob. Set 1.1); so (b) is the envelope of (a). Such a solution (b) that cannot be obtained from a general solution is called a singular solution.

If a body slides on a surface, it experiences friction F (a force against the direction of motion). Experiments show that |F| = Î¼|N| (Coulomb's law of kinetic friction without lubrication), where N is the normal force (force that holds the two surfaces together; see Fig. 15) and the constant of proportionality Î¼ is called the coefficient of kinetic friction. In Fig. 15 assume that the body weighs 45 nt (about 10 lb; see front cover for conversion). Î¼ = 0.20 (corresponding to steel on steel), a = 30Â° the slide is 10 m long, the initial velocity is zero, and air resistance is negligible. Find the velocity of the body at the end of the slide.

Suppose that the population y(t) of a certain kind of fish is given by the logistic equation (11), and fish are caught at a rate Hy proportional to y. Solve this so-called Schaefer model. Find the equilibrium solutions y_{1} and y_{2 }(> 0) when H < A. The expression Y = H_{y2 }is called the equilibrium harvest or sustainable yield corresponding to H. Why?

In Prob. 36 assume that you fish for 3 years, then fishing is banned for the next 3 years. Thereafter you start again. And so on. This is called intermittent harvesting. Describe qualitatively how the population will develop if intermitting is continued periodically. Find and graph the solution for the first 9 years, assuming that A = B = 1, H = 0.2, and y(0) = 2.

**Data from Prob. 36**

_{1} and y_{2 }(> 0) when H < A. The expression Y = H_{y2 }is called the equilibrium harvest or sustainable yield corresponding to H. Why?

Apply the given operator to the given functions. Show all steps in detail.

D - 3I; 3x^{2} + 3x, 3e^{3x},^{ }cos 4x - sin 4x

Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9.

y" + 36y = 0

Apply the given operator to the given functions. Show all steps in detail.

(D + 6I)^{2}; 6x + sin 6x, xe^{-6x}

Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9.

y" + 4y' + (π^{2} + 4)y = 0

Find the steady-state motion of the mass–spring system modeled by the ODE. Show the details of your work.

y" + 2.5y' + 10y = -13.6 sin 4t

Reduce to first order and solve, showing each step in detail.

2xy'' = 3y'

Reduce to first order and solve, showing each step in detail.

xy'' + 2y' + xy = 0, y_{1} = (cos x)/x

Apply the given operator to the given functions. Show all steps in detail.

(D^{2} + 4.00D + 3.36I)y = 0

Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9.

10y" - 32y' + 25.6y = 0

Apply the given operator to the given functions. Show all steps in detail.

(D^{2} + 3I)y = 0

y" + y' + 3.25y = 0

Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.

2y" + 4y' + 6.5y = 4 sin 1.5t

Reduce to first order and solve, showing each step in detail.

y'' = 1 + y'^{2}

Apply the given operator to the given functions. Show all steps in detail.

(D2 + 4.80D + 5.76I)y = 0

100y" + 240y' + (196π^{2} + 144)y = 0

Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.

y" + 16y = 56 cos 4t

Reduce to first order and solve, showing each step in detail.

y'' + (1 + 1/y)y'^{2 }=^{ }0

Apply the given operator to the given functions. Show all steps in detail.

(D^{2} + 3.0D + 2.5I)y = 0

y" + 9y' + 20y = 0

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