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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

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" + 36y = 0
Apply the given operator to the given functions. Show all steps in detail.D - 3I; 3x2 + 3x, 3e3x, cos 4x - sin 4x
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
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
If a body slides on a surface, it experiences friction F (a force against the direction of motion). Experiments show that |F| = μ|N| (Coulomb's law of kinetic friction without lubrication), where N
A Riccati equation is of the form? ?y' + p(x)y = g(x)y2 + 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
A rocket is shot straight up from the earth, with a net acceleration (= acceleration by the rocket engine minus gravitational pullback) of 7t m/sec2 during the initial stage of flight until the
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
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.2xyy' + (x - 1)y2 =
Could you see, practically without calculation, that the answer in Prob. 27 must lie between 60 and 70 min? Explain.Data from Prob 27If a wet sheet in a dryer loses its moisture at a rate
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
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),
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' + y = -x/y
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
Solve the IVP. Indicate the method used. Show the details of your work.y' + 1/2y = y3, y(0) = 1/3
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 = y2, y(0) = -1/3
Solve the IVP. Indicate the method used. Show the details of your work.y' + 4xy = e-2x2, y(0) = -4.3
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
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
(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
Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.y' = ay + by2 (a ≠ 0)
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,
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
Radium 22488 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?
Introduce limits of integration in (3) such that y obtained from (3) satisfies the initial condition y(x0) = y0.
Graph particular solutions of the following ODE, proceeding as explained.(21)? ? ?dy - y2sin x dx = 0 (a) Show that (21) is not exact. Find an integrating factor using either Theorem 1 or 2. Solve
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
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,
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
If y' = f(x) with f independent of y, show that the curves of the corresponding family are congruent, and so are their OTs.
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
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)
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 - y2, y(0) = 0.2 by Euler’s method
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
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
(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
Find the conditions under which the orthogonal trajectories of families of ellipses x2/a2 + y2/b2 = c are again conic sections. Illustrate your result graphically by sketches or by using your CAS.
Solve the IVP. Show the steps of derivation, beginning with the general solution.dr/dt = -2tr, r(0) = r0
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
Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare.xy' = y + x2
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
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.)xy' + 4y = 8x4, y(1) = 2
(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, y2 - 4x2 = c (y > 0), y(1) = 4
Solve the IVP. Show the steps of derivation, beginning with the general solution.y' = 1 + 4y2, y(1) = 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
Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare.y' = 1 - y2
Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.x2 + (y - c)2 = c2
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' cos x + (3y - 1) sec x = 0,
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 sin2 x + 4, y(1/2π) = 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.y' + 5xy = 0, y = ce-2.5x2, y(0) = π
Find all initial conditions such that (x2 - x)y' = (2x - 1)y has no solution, precisely one solution, and more than one solution.
Find a general solution. Show the steps of derivation. Check your answer by substitution.xy' = x + y (Set y/x = u)
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
Does the initial value problem (x - 2)y' = y, y(2) = 1 have a solution? Does your result contradict our present theorems?
Solve the ODE by integration or by remembering a differentiation formula.y' + xe-x2/2 = 0
Find a general solution. Show the steps of derivation. Check your answer by substitution.y3y' + x3 = 0
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 = x3 and passing through the origin (0, 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).yy' + 4x = 0, (1, 1), (2, 1/2)
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?
Find a general solution. Show the steps of derivation. Check your answer by substitution.y' sin 2πx = πy cos 2πx
Solve the ODE by integration or by remembering a differentiation formula.y' = -1.5y
Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.y = x2 + c
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
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 - y2, (0, 0), (0, 1), (0, 2), (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
(a) Apply the iteration to y' = x + y, y(0) = 0. Also solve the problem exactly.(b) Apply the iteration to y' = 2y2, y(0) = 1. Also solve the problem exactly.(c) Find all solutions of y' = 2√y,
Find a general solution. Show the steps of derivation. Check your answer by substitution.y' = e2x-1 y2
Solve the ODE by integration or by remembering a differentiation formula.y'' = -y
Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.xy = c
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
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' = sin2 y, (0, -0.4), (0, 1)
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
Find a general solution. Show the steps of derivation. Check your answer by substitution.y' = (y + 4x)2 (Set y + 4x = v)
Show that for a linear ODE y' + p(x)y = r(x) with continuous p and r in |x - x0| < a Lipschitz condition holds. This is remarkable because it means that for a linear ODE the continuity of f(x, y)
Solve the ODE by integration or by remembering a differentiation formula.y''' = e-0.2x
Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.y = √x + c
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)
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)
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
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
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.
Suppose that the tank in Example 7 is hemispherical, of radius R, initially full of water, and has an outlet of 5 cm2 cross sectional area at the bottom. (Make a sketch.) Set up the model for

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