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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

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
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 +
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
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
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
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
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
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
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
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
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
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
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
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
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
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
(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
(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
(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
(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,
(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
(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)
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
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 =
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
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
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
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
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
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
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
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,
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
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
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')
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
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
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
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 —
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
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
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
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
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'
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
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
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
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
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
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
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

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