New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
mathematics
first course differential equations
A First Course in Differential Equations with Modeling Applications 11th edition Dennis G. Zill - Solutions
What is the slope of the tangent line to the graph of a solution y' = 6√(y + 5x3) of that passes through (1, 4)?
Verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution.1. y'' + y = 2 cos x 2 sin x; y = x sin x + x cos x2. y'' + y = sec x; y = x sin x + (cos x)ln(cos x)3. x2y'' + xy' + y = 0; y = sin(ln x)4. x2y'' +
Verify that the indicated expression is an implicit solution of the given differential equation.1. x dy/dx + y = 1/y2 ; x3y3 = x3 + 12. (dy/dx)2 + 1 = 1/y2 ; (x - 5)2 + y2 = 13. y'' = 26(y')3 ; y3 + 3y = 1 – 3x4. (1 = xy)y' = y2 ; y = exy
A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water and water leaks from a circular hole of radius 1/2 inch at its bottom, determine a differential equation for the height h of the water at time t > 0.
In Problems y = c1e3x + c2ex 2x is a two parameter family of the second-order DE y'' 2y' 3y = 6x + 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.1. y (0) = 0, y'(0) = 02. y (0) = 1, y'(0) = 33. y (1) = 4, y'(1) = -24. y
The graph of a solution of a second-order initial-value problem d2y/dx2=Â f (x, y, y'), y(2) =Â y0, y'(2) =Â y1, is given in the following figure. Use the graph to estimate the values of y0and y1. Ун
At a time denoted as t = 0 a technological innovation is introduced into a community that has a fixed population of n people. Determine a differential equation for the number of people x(t) who have adopted the innovation at time t if it is assumed that the rate at which the innovations spread
Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. Determine a differential equation for the number of people x(t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between the number of
The ambient temperature Tm in (3) could be a function of time t. Suppose that in an artificially controlled environment, Tm(t) is periodic with a 24-hour period, as illustrated in the following figure. Devise a mathematical model for the temperature T(t) of a body within this environment.
A cup of coffee cools according to Newton€™s law of cooling (3). Use data from the graph of the temperature T(t) in the following figure to estimate the constants Tm, T0, and k in a model of the form of a first-order initial-value problem: dT/dt = k(T Tm), T(0) = T0. т 200
Modify the model in Problem 3 for net rate at which the population P(t) of a certain kind of fish changes by also assuming that the fish are harvested at a constant rate h > 0.
Using the concept of net rate introduced in Problem 2, determine a model for a population P(t) if the birth rate is proportional to the population present at time t but the death rate is proportional to the square of the population present at time t.
The population model given in (1) fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a community it is assumed that the rate at which the population changes is a net rate that is, the difference between the rate of births and
The functions y(x) = 1/16 x4, -ˆž < x < ˆž and have the same domain but are clearly different. See the following figure (a) and (b), respectively. Show that both functions are solutions of the initial-value problem dy/dx = xy1/2, y(2) = 1 on the interval
Suppose that the first-order differential equation dy/dx = f (x, y) possesses a one-parameter family of solutions and that f (x, y) satisfies the hypotheses of Theorem 1.2.1 in some rectangular region R of the xy-plane. Explain why two different solution curves cannot intersect or be tangent to
Determine a plausible value of x0 for which the graph of the solution of the initial-value problem y' + 2y = 3x 6, y(x0) = 0 is tangent to the x-axis at (x0, 0). Explain your reasoning.
Consider the initial-value problem y' = x 2y, y(0) = 1/2. Determine which of the two curves shown in following figure is the only plausible solution curve. Explain your reasoning. х 1.
Use Problem 55 in Exercises 1.1 and (2) and (3) of this section.1. Find a function y = f (x) whose graph at each point (x, y) has the slope given by 8e2x + 6x and has the y-intercept (0, 9).2. Find a function y f (x) whose second derivative is y'' = 12x 2 at each point (x, y) on its graph
In Problems y = c1 cos 2x + c2 sin 2x is a two paramet4er family of solutions of the second order DE y'' + 4y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions The condition specified at two different points are called boundary conditions.1.
The graph of a member of a family of solutions of a second-order differential equation d2y/dx2= f (x, y, y') is given. Match the solution curve with at least one pair of the following initial conditions.(a) y(1) = 1, y'(1) = -2(b) y(-1) = 0, y'(1) = -4(c) y(1) = 1, y' (1) = 2(d) y(0) = -1, y' (0) =
(a) Use the family of solutions in part (a) of Problem 33 to find an implicit solution of the initial value problem y dy/dx = 3x, y(2)=4. Then, by hand, sketch the graph of the explicit solution of this problem and give its interval I of definition(b) Are there any explicit solutions of y dy/dx =
(a) Show that a solution from the family in part (a) of Problem 31 that satisfies y' = y2, y(1) = 1, is y = 1/(2 x).(b) Then show that a solution from the family in part (a) of Problem 31 that satisfies y' = y2, y(3)= 1, is y = 1/(2 x).(c) Are the solutions in parts (a) and (b) the same?
(a) Verify that 3x2 y2 = c is a one-parameter family of solutions of the differential equation y dy/dx = 3x.(b) By hand, sketch the graph of the implicit solution 3x2 y2 = 3. Find all explicit solutions y = ϕ(x) of the DE in part (a) defined by this relation. Give the interval I of definition
(a) Verify that y = -1 / (x + c) is a one-parameter family of solutions of the differential equation y' = y2.(b) Since f (x, y) = y2 and Ï‘f/Ï‘y = 2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Find a solution from the family
(a) Verify that y = tan (x + c) is a one-parameter family of solutions of the differential equation y' = 1 + y2.(b) Since f (x, y) = 1 + y2 and ϑf/ϑy = 2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to
(a) By inspection find a one-parameter family of solutions of the differential equation xy' = y. Verify that each member of the family is a solution of the initial-value problem xy' = y, y(0) = 0.(b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy'
Determine whether Theorem 1.2.1 guarantees that the differential equation y' = ˆš(y2- 9) possesses a unique solution through the given point.1. (1, 4)2. (5, 3)3. (2, 3)4. (1, 1)Theorem 1.2.1Let R be a rectangular region in the xy-plane defined by a ‰¤ x ‰¤ b, c
Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the1. dy/dx = y2/32. dy/dx = √xy3. x dy/dx = y4. dy/dx –y = x5. (4 – y2)y' = x26. (1 + y3)y' = x27. (x2 + y2)y' = y28. (y - x)y' = y + x
Determine by inspection at least two solutions of the given first-order IV .1. y' = 3y2/3, y(0) = 02. xy' = 2y, y(0) = 0
In Problems y = c1ex + c2e-x is a two-parameter family of solutions of the second-order DE y'' -y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.1. y(0) = 1, y'(0) = 22. y(1) = 0, y'(1) = e3. y(-1) = 5, y'(-1) = -54. y(0) = 0,
In Problems x = c1 cos t + c2 sin t is a two-parameter family of solutions of the second order DE '' + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.1. x(0) = -1, x'(0) = 82. x(π/2) = 0, x'(π/2) = 13. x(π/6) = ½,
In Problems y = 1/(x2 + c) is a one-parameter family of solutions of the first-order DE y' + 2xy2 = 0. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined1. y(2) = 1/32. y(-2)
In Problems y = 1/(1+ c1e-x) is a one-parameter family of solutions of the first order DE y' = y -y2. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.1. y(0) = -1/32. y(1) = 2
In Problems 63 and 64 use a CAS to compute all derivatives and to carry out the simplifications needed to verify that the indicated function is a particular solution of the given differential equation.y(4) −20y''' + 158y'' − 580y' + 841y = 0;y = xe5x cos 2xx3y''' + 2x2y'' + 20xy' – 78y =
Consider the differential equation y' = y2 + 4.(a) Explain why there exist no constant solutions of the DE.(b) Describe the graph of a solution y = ϕ(x). For example, can a solution curve have any relative extrema?(c) Explain why y = 0 is the y-coordinate of a point of inflection of a solution
Consider the differential equation dy/dx = y(a by), where a and b are positive constants.(a) Either by inspection or by the method suggested in Problems 37– 40, find two constant solutions of the DE.(b) Using only the differential equation, find intervals on the y-axis on which a nonconstant
Consider the differential equation dy/dx = 5 y.(a) Either by inspection or by the method suggested in Problems 33– 36, find a constant solution of the DE.(b) Using only the differential equation, find intervals on the y-axis on which a non constant solution y = ϕ(x) is increasing. Find
Consider the differential equation dy/dx = e-x.(a) Explain why a solution of the DE must be an increasing function on any interval of the x- axis.(b) What are What does this suggest about a solution curve as x †’ ± ˆž(c) Determine an interval over which a
Find a linear second-order differential equation F(x, y, y', y'') = 0 for which y = c1x + c2x2 is a two parameter family of solutions. Make sure that your equation is free of the arbitrary parameters c1 and c2.Qualitative information about a solution y = φ(x) of a differential equation can often
The normal form (5) of an nth-order differential equation is equivalent to (4) whenever both forms have exactly the same solutions. Make up a first-order differential equation for which F(x, y, y') 0 is not equivalent to the normal form dy/dx = f (x, y).
The differential equation x(y')2 4y' 12x3 0 has the form given in (4). Determine whether the equation can be put into the normal form dy/dx = f (x, y).
Discuss, and illustrate with examples, how to solve differential equations of the forms dy/dx = f (x) and d2y/dx2 = f (x).
In Problem 21 a one-parameter family of solutions of the DE P' = P(1 P) is given. Does any solution curve pass through the point (0, 3)? Through the point (0, 1)?Data from problem 21Verify that the indicated family of functions is a solution of the given differential equation. Assume an
In Example 5 the largest interval I over which the explicit solutions y = ϕ1(x) and y = ϕ2(x) are define is the open interval (5, 5). Why can’t the interval I of definition be the closed interval 5, 5]?
The following graph is the member of the family of folia in Problem 51 corresponding to c = 1. Discuss: How can the DE in Problem 51 help in finding points on the graph of x3+ y3= 3xy where the tangent line is vertical? How does knowing where a tangent line is vertical help in determining an
The graphs of members of the one-parameter family x3+ y3= 3cxy are called folia of Descartes. Verify that this family is an implicit solution of the first-orde differential equation У(у — 2х3) dy x(2y³ – x³)
The given figure represents the graph of an implicit solution G(x, y) = 0 of a differential equation dy/dx = f (x, y). In each case the relation G(x, y) = 0 implicitly defines several solutions of the DE. Carefully reproduce each figure on a piece of paper. Use different colored pencils to mark off
Discuss why it makes intuitive sense to presume that the linear differential equation y'' + 2y' + 4y = 5 sin t has a solution of the form y = A sin t + B cos t, where A and B are constants. Then find specific constants A and B so that y = A sin t + B cos t is a particular solution of the DE.
Given that y = sin x is an explicit solution of the first order differential equation dy/dt = √ (1 – y2) Find an interval I of definition.
What function (or functions) do you know from calculus is such that its second derivative is itself? Its second derivative is the negative of itself? Write each answer in the form of a second-order differential equation with a solution.
What function do you know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple k of itself? Write each answer in the form of a first-order differential equation with a solution.
Make up a differential equation that you feel confident possesses only the trivial solution y = 0. Explain your reasoning.
Use the concept that y = c, -∞ < x < ∞, is a constant function if and only if y' = 0 to determine whether the given differential equation possesses constant solutions.1. 3xy' + 5y = 102. y' = y2 + 2y - 33. (y - 1)y' = 14. y'' + 4y' + 6y = 10
In Problems 1–4 find values of m so that the function y =emx is a solution of the given differential equation.1. y' + 2y = 02. 5y' = 2y3. y'' - 5y' + 6y = 04. 2y'' + 7y' - 4y = 0
Discuss why it is technically incorrect to say that the function in (10) is a “solution” of the IVP on the interval [0, ∞).
Construct a linear first order differential equation for which all solutions are asymptotic to the line y = 3x - 5 as x→ ∞.
Find the general solution of the differential equation on the interval (-3, 3).
Reread Example 3 and then discuss, with reference to Theorem 1.2.1, the existence and uniqueness of a solution of the initial-value problem consisting of xy' - 4y = x6ex and the given initial condition.(a) y(0) = 0(b) y(0) = y0, y0 > 0(c) y(x0) = y0, x0 . 0, y0 > 0Theorem 1.2.1Let R be a
Construct a linear first-order differential equation for which all non constant solutions approach the horizontal asymptote y = 4 as x → ∞.
The Fresnel sine integral function is defined asSee Appendix A. Express the solution of the initial-value problem dy/dx - (sinx2)y = 0, y(0) = 5 in terms of S(x). dt. sin S(x) = sın
The sine integral function is defined aswhere the integrand is defined to be 1 at x = 0. See Appendix A. Express the solution of the initial-value problem x3 dy/dx - 2x2y = 10sinx, y(1) = 0 in terms of Si(x). sint Si(x) = dt,
express the solution of the given initial-value problem in terms of an integral defined function.x2 dy/dx - y = x3, y(1) = 0
Express the solution of the given initial-value problem in terms of an integral defined function.dy/dx + exy = 1, y(0) = 1
Express the solution of the given initial-value problem in terms of an integral defined function.dy/dx + exy = 1, y(0) = 1
Use a graphing utility to graph the continuous function y(x).dy/dx + P(x)y = 0, y(0) = 4, where 1, 0sxs2 P(x) = 5,
Use a graphing utility to graph the continuous function y(x).dy/dx + P(x)y = 4x, y(0) = 3, where 2, 0
Proceed as in Example 6 to solve the given initial value problem. Use a graphing utility to graph the continuous function y(x).(1 + x2) dy/dx + 2xy = f (x), y(0) = 0, where X, f(x) = -x, -x,
Use a graphing utility to graph the continuous function y(x).(1 + x2) dy/dx + 2xy = f (x), y(0) = 0, where X, f(x) = -x, -x,
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution.y dx = (yey - 2x) dy
Express the solution of the given initial-value problem in terms of erfc(x).dy/dx - 2xy = 1, y(1) = 1
Express the solution of the given initial-value problem in terms of erfc(x).dy/dx - 2xy = -1, y(0) = √π/y2
Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. Find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare
Show that an implicit solution of2x sin2 y dx - (x2 + 10) cos y dy = 0is given by ln(x2 + 10) 1 csc y = c. Find the constant solutions, if any, that were lost in the solution of the differential equation.
Express the solution of the given initial-value problem in terms of an integral defined function.x dy/dx + 2y = xex2, y(1) = 3
Fill in the blanks or answer true or false.An example of an autonomous linear first-order DE with a single critical point -3 is ________, whereas an autonomous nonlinear first-order DE with a single critical point -3 is ________.
Solve the given differential equation by using an appropriate substitution.dy/dx = (1 - x – y)/( x + y)
Solve the given differential equation by using an appropriate substitution.dy/dx = (x + y + 1)2
Solve the given initial-value problem.y1/2 dy/dx + y3/2 = 1, y(0) = 4
Solve the given initial-value problem.x2 dy/dx - 2xy = 3y4, y(1) = 1/2
Solve the given differential equation by using an appropriate substitution.3(1 + t2) dy/dt = 2ty(y3 - 1)
Solve the given differential equation by using an appropriate substitution.t2 dy/dt + y2 = ty
Solve the given differential equation by using an appropriate substitution.x dy/dx - (1 + x)y = xy2
Solve the given differential equation by using an appropriate substitution.dy/dx = y(xy3 - 1)
Solve the given differential equation by using an appropriate substitution.dy/dx - y = exy2
Solve the given differential equation by using an appropriate substitution.x dy/dx + y = 1/y2
Solve the given initial-value problem.y dx + x(ln x - ln y - 1) dy = 0, y(1) = e
Solve the given initial-value problem.(x + yey/x) dx - xey/x dy = 0, y(1) = 0
Solve the given initial-value problem.(x2 + 2y2) dx/dy = xy, y(-1) = 1
Solve the given initial-value problem.xy2 dy/dx = y3 - x3, y(1) = 2
Solve the given differential equation by using an appropriate substitution.x dy/dx = y + √(x2 - y2), x > 0
Solve the given differential equation by using an appropriate substitution.-y dx + (x + √xy + dy = 0
Solve the given differential equation by using an appropriate substitution.dy/dx = x + 3y/3x + y
Solve the given differential equation by using an appropriate substitution.dy/dx = y – x/y + x
Solve the given differential equation by using an appropriate substitution.(y2 + yx) dx + x2 dy = 0
Solve the given differential equation by using an appropriate substitution.(y2 + yx) dx - x2 dy = 0
Solve the given differential equation by using an appropriate substitution.y dx = 2(x + y) dy
Solve the given differential equation by using an appropriate substitution.x dx + (y - 2x) dy = 0
Solve the given differential equation by using an appropriate substitution.(x + y) dx + x dy = 0
Solve the given differential equation by using an appropriate substitution.(x - y) dx + x dy = 0
(a) The solution of the differential equation{2xy/(x2 + y2)2 [1 + (y2 – x2)/(x2 + y2)2]dy = 0is a family of curves that can be interpreted as streamlines of a fluid flow around a circular object whose boundary is described by the equation x2 + y2 = 1. Solve this DE and note the solution f (x, y)
Showing 2200 - 2300
of 2513
First
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Step by Step Answers
Get 50% OFF
Claim Now
