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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Reduce to first order and solve, showing each step in detail.y'' + y' = 0
By what methods can you get a general solution of a nonhomogeneous ODE from a general solution of a homogeneous one?
How does the frequency of the harmonic oscillation change if we (i) double the mass, (ii) take a spring of twice the modulus? First find qualitative answers by physics, then look at formulas.
Find the steady-state motion of the mass–spring system modeled by the ODE. Show the details of your work.y" + 6y' + 8y = 42.5 cos 2t
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.x2y" – 2xy' + 2y = x3 sin x
Find a (real) general solution. State which rule you are using. Show each step of your work.y" + 3y' + 2y = 12x2
Apply the given operator to the given functions. Show all steps in detail.(D - 2I)2; e2x, xe2x, e-2x
Find a real general solution. Show the details of your work.5x2y" + 23xy' + 16.2y = 0
What does an initial value problem of a second-order ODE look like? Why must you have a general solution to solve it?
Show that F (x, y', y'') = 0 can be reduced to first order in z = y' (from which y follows by integration). Give two examples of your own.
Why are linear ODEs preferable to nonlinear ones in modeling? What does an initial value problem of a second-order ODE look like? Why must you have a general solution to solve it?
Find the harmonic motion (4) that starts from y0 with initial velocity v0. Graph or sketch the solutions for ω0 = π, y0 = 1, and various v0 of your choice on common axes. At what t-values
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 2.9.4y" - 25y = 0
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.y" + 9y = sec 3x
Find a (real) general solution. State which rule you are using. Show each step of your work.y" + 5y' + 4y = 10e-3x
Apply the given operator to the given functions. Show all steps in detail.D2 + 2D; cosh 2x, e-x + e2x, cos x
Verify directly by substitution that x(1-a)/2 ln x is a solution of (1) if (2) has a double root, but xm1 and xm2 ln x are not solutions of (1) if the roots m1 and m2 of (2) are
If in a population y(t) the death rate is proportional to the population, and the birth rate is proportional to the chance encounters of meeting mates for reproduction, what will the model be?
In Prob. 36 find and graph the solution satisfying y(0) = 2 when (for simplicity) A = B = 1 and H = 0.2. What is the limit? What does it mean? What if there were no fishing?Data from Prob. 36Suppose
Lake Erie has a water volume of about 450 km3 and a flow rate (in and out) of about 175 km2 per year. If at some instant the lake has pollution concentration p = 0.04%, how long, approximately,
A CAS can usually graph solutions, even if they are integrals that cannot be evaluated by the usual analytical methods of calculus.(a) Show this for the five initial value problems y' = e-x2, y(0) =
Find and solve the model for drug injection into the bloodstream if, beginning at t = 0, a constant amount A g/min is injected and the drug is simultaneously removed at a rate proportional to the
If the temperature of a cake is 300°F when it leaves the oven and is 200°F ten minutes later, when will it be practically equal to the room temperature of 60°F, say, when will it be 61°F?
We have transformed ODEs to separable form, to exact form, and to linear form. The purpose of such transformations is an extension of solution methods to larger classes of ODEs. Describe the key idea
If in a reactor, uranium 23797 U loses 10% of its weight within one day, what is its half-life? How long would it take for 99% of the original amount to disappear?
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' = 1/(6ey - 2x)
If the growth rate of a culture of bacteria is proportional to the number of bacteria present and after 1 day is 1.25 times the original number, within what interval of time will the number of
Solve the IVP. Indicate the method used. Show the details of your work.3 sec y dx + 1/3 sec x dy = 0, y(0) = 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' + xy = xy-1y(0) = 3
Experiments show for a gas at low pressure p (and constant temperature) the rate of change of the volume V(p) equals -V/p. Solve the model.
Solve the IVP. Indicate the method used. Show the details of your work.y' = √1 - y2y(0) = 1/√2
Another method of obtaining (4) results from the following idea. Write (3) as cy*where y* is the exponential function, which is a solution of the homogeneous linear ODE y*' + py* = 0.
What should be the 146C content (in percent of y0) of a fossilized tree that is claimed to be 3000 years old?
Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.(3xey + 2y) dx + (x2ey + x) dy = 0
In dropping a stone or an iron ball, air resistance is practically negligible. Experiments show that the acceleration of the motion is constant (equal to g = 9.80 m/sec2 = 32 ft/sec2 called the
If the growth rate of the number of bacteria at any time t is proportional to the number present at t and doubles in 1 week, how many bacteria can be expected after 2 weeks? After 4 weeks?
Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.25yy' - 4x = 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,
Solve the IVP. Show the steps of derivation, beginning with the general solution.xy' = y + 3x4 cos2 (y/x)y(1) = 0(Set y/x = u)
Working backward from the solution to the problem is useful in many areas. Euler, Lagrange, and other great masters did it. To get additional insight into the idea of integrating factors, start from
Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.y' + 2.5y = 1.6x
Show that for a family u(x, y) = c = const the orthogonal trajectories v(x, y) = c* = const can be obtained from the following Cauchy–Riemann equations (which are basic in complex analysis in Chap.
Solve the IVP. Show the steps of derivation, beginning with the general solution.y' = -4x/yy(2) = 3
Under what conditions for the constants a, b, k, l is (ax + by) dx + (kx + ly) dy = 0 exact? Solve the exact ODE.
Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare.y' + y = 1.01 cos 10x
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.)y' = 6(y - 2.5) tanh 1.5x
Let the isotherms (curves of constant temperature) in a body in the upper half-plane y > 0 be given by 4x2 + 9y2 = c. Find the orthogonal trajectories (the curves along which heat will flow in
(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' = y - y2y = 1/1 + ce-xy(0) = 0.25
Solve the IVP. Show the steps of derivation, beginning with the general solution.y' cosh2 x = sin2 yy(0) = 1/2π
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' = y - 4y2
This means an ODE not showing x (the independent variable) explicitly. What will the level curves f (x, y) = const (also called isoclines = curves of equal inclination) of an autonomous ODE look
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 - 2) cot x
Let the electric equipotential lines (curves of constant potential) between two concentric cylinders with the z-axis in space be given by u(x, y) = x2 + y2 = c (these are circular cylinders in the
(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' = y + exy = (x + c)exy(0) = 1/2
Solve the IVP. Show the steps of derivation, beginning with the general solution.xy' + y = 0 y(4) = 6
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' + 2y = 0
What is the largest possible ? in Example 1 in the text?Data from example 1Consider the initial value problemand take the rectangle R; |x| This solution is discontinuous at + ?/2, and there is no
Direction fields are very useful because they can give you an impression of all solutions without solving the ODE, which may be difficult or even impossible. To get a feel for the accuracy of the
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 sin x = ecos xy(0) = -2.5
Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.y = ce-x2
(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' + 4y = 1.4y = ce-4x + 0.35y(0) = 2
Can two solution curves of the same ODE have a common point in a rectangle in which the assumptions of the present theorems are satisfied?
Find a general solution. Show the steps of derivation. Check your answer by substitution.xy' = y2 + 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
Give problems from mechanics, heat conduction, and population dynamics that can be modeled by first-order ODEs.
What does modeling mean? Can a CAS solve a model given by a first-order ODE? Can a CAS set up a model?
Can an ODE sometimes be solved by several methods? Give three examples.
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' = ey/x(2, 2), (3, 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.)xy' = 2y + x3ex
Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.y = c/x2
Solve the ODE by integration or by remembering a differentiation formula.y' = cosh 5.13x
Find a general solution. Show the steps of derivation. Check your answer by substitution.xy' = y + 2x3 sin2 y/x (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
Explain the idea of an integrating factor. Give two examples.
What is an exact ODE? Is f(x) dx + g(y) dy = 0 always exact?
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' = x - 1/y(1, 1/2)
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' + ky = e-kx
Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.y = cx
Solve the ODE by integration or by remembering a differentiation formula.y' = 4e-x cos x
Find a general solution. Show the steps of derivation. Check your answer by substitution.yy' + 36x = 0
In most cases the solution of an initial value problem (1) exists in an x-interval larger than that guaranteed by the present theorems. Show this fact for y' = 2y2, y(1) = 1 by finding the best
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
What is a direction field? A numeric method for first order ODEs?
Does every first-order ODE have a solution? A solution formula? Give examples.
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' = 1 - y2(0, 0), (2, 1/2)
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 = 5.2
Represent the given family of curves in the form G(x, y; c) = 0 and sketch some of the curves.The catenaries obtained by translating the catenary y = cosh x in the direction of the straight line y =
If the assumptions of Theorems 1 and 2 are satisfied not merely in a rectangle but in a vertical infinite strip |x - x0| < a in what interval will the solution of (1) exist?
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
What is a linear ODE? Why is it easier to solve than a nonlinear ODE?
Give a reason why in (4) you may choose the constant of integration in ∫p dx to be zero.
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' = 1 + y2(1,1), |(0,1/2)
Show that e-ln x = 1/x (not -x) and e-ln(sec x) = cos x.
Represent the given family of curves in the form G(x, y; c) = 0 and sketch some of the curves.All ellipses with foci -3 and 3 on the x-axis.
Why is it important to introduce the constant of integration immediately when you integrate?
Solve the ODE by integration or by remembering a differentiation formula.y' + 2 sin 2πx = 0
If p and r in are y' + p(x)y = r(x) are continuous for all x in an interval |x - x0| < show that in this ODE satisfies the conditions of our present theorems, so that a corresponding initial value

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