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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

A tightly stretched, flexible, uniform string has its ends fixed at the points x = 0 and x = l. The midpoint of the string is displaced a distance a, as shown in Figure 12.22. If f(x) denotes the
Repeat Question 18 for the case where the displaced profile of the string is as shown in Figure 12.23.Data from Question 18A tightly stretched, flexible, uniform string has its ends fixed at the
function f(t) is defined on 0 ≤ t ≤ π byFind a half-range Fourier series expansion of f(t) on this interval. Sketch a graph of the function represented by the series for –2π ≤ t ≤ 2π.
A function f(t) is defined on the interval –l ≤ x ≤ l byObtain a Fourier series expansion of f (x) and sketch a graph of the function represented by the series for –3l ≤ x ≤ 3l. A f(x) =
The temperature distribution T(x) at a distance x, measured from one end, along a bar of length L is given byExpress T(x) as a Fourier series expansion consistingof sine terms only. T(x) = Kx(Lx)
Find the Fourier series expansion of the function f(t) valid for –1 To what value does this series converge when t = 1? f(t) = 1 ost (-1 < t < 0) (0
Show that the periodic functionhas a Fourier series expansionBy term-by-term integration of this series, show that the periodic function f(t) = t (-T
The periodic functionhas a Fourier series expansionBy term-by-term differentiation of this series, confirm the series obtained for f(t) in Question 24 for the case when T = π.Data from Question
(a) In Example 12.7 we saw that the periodic functionhas a Fourier series expansionDifferentiate this series term by term, and explain why it is not a Fourier expansion of the periodic function(b)
A periodic function f(t) is defined byObtain a Fourier series expansion of f(t) and deduce that f(t) = {. (0 t < ) (
Determine the full-range Fourier series expansion of the even function f(t) of period 2π defined byTo what value does the series converge at t = 1/3π? f(t) = (1) - (0 t n) (1 T)
A function f(t) is defined for 0 ≤ t ≤ 1/2T byf(t)={t(0⩽t⩽14T)12T−t(14T⩽t⩽12T)f(t)={t(0⩽t⩽14T)12T−t(14T⩽t⩽12TSketch odd and even functions that have a period T and are equal
The magnetomotive force, y, in the air gap of an alternator can be represented approximately by a graph of the form shown in Figure 12.25. Find a Fourier series for y, explaining beforehand, with
Prove that if g(x) is an odd function and f(x) an even function of x, the product g(x)[c + f(x)] is an odd function if c is a constant. A periodic function with period 2π is defined byin the
A repeating waveform of period 2π is described bySketch the waveform over the range t = –2π to t = 2 and find the Fourier series representation of f(t), making use of any properties of the
A function f(x) is periodic of period 2 and is defined bySketch a graph of f(x) from –2π to 3π and prove thatHence show that f(x) = [-2x (- < x 0) 2x (0 < x )
A function f(x) of period 2π is defined in the interval –π ≤ x ≤ π bySketch a graph of f(x) over the interval –3π ≤ x ≤ 3π. Express f(x) as a Fourier series and from this deduce a
A periodic function of period 2π is defined for 0 ≤ x ≤ 2π bySketch f(x) for –2π ≤ x ≤ 4π and show that its Fourier series representation isExpress this series in a general form. f(x) =
A waveform is defined by V(t) = 10e–3t for 0 ≤ t < 0.4 and V(t) = V(t – 0.4) for all t. Sketch the graphs of V, dV/dt and ∫t0 Vdt. Express V as a Fourier series and show that the amplitude of
A function f(x) is defined in the interval –1 ≤ x ≤ 1 bySketch a graph of f(x) and show that a Fourier series expansion of f(x) valid in the interval–1 ≤ x ≤ 1 is given by f(x) = [1/28 (-e
Show that the half-range Fourier sine series for the function is 500)=(1-2)* f(t) (0 t ) 50) = { 1 - 201- (-101} sin er f(t n n n=1
Find a half-range Fourier sine and Fourier cosine series for f (x) valid in the interval 0 Sketch the graph of the Fourier series obtained for –2π, x ≤ 2π. X (0 x = n) f(x) = -- n-x x ) X
A function f(x) is periodic of period 2π and is defined by f(x) = ex (–π f(x) = 2 sinh (-1)" + (cos nx - n sin nx) 1 + n2 n=1
A function f(t) is defined on 0 Find(a) A half-range Fourier sine series, and(b) A half-range Fourier cosine series for f(t) valid for 0 f(t) = - t
A periodic function f(t) of period 2 is defined in the interval –1 < t < 1 by f(t) = 1 – t2 Sketch a graph of f (t) for –3 < t < 3 and obtain a Fourier series expansion for it.
(a) Without actually finding the series, state what terms you would expect to find in the Fourier series for the following periodic functions of period 2π:(b) Find, up to and including the term in
(a) A periodic function f(t), of period 2π, is defined in –π ≤ t ≤ π byObtain a Fourier series expansion for f(t).(b) By formally differentiating the series obtained in (a), obtain the
The periodic waveform f(t) shown in Figure 12.26 may be written aswhere g(t) represents an odd function.(a) Sketch the graph of g(t).(b) Obtain the Fourier series expansion for g(t), and hence write
Show that the Fourier series represents the function f(t), of period 2π, given byDeduce that, apart from a transient component (that is, a complementary function that dies away as t → ∞), the
Show that if f(t) is a periodic function of period 2π andShow also that, when ω is not an integer,satisfies the differential equationsubject to the initial conditions y = dy/dt = 0 at t = 0. then
Classify each of the following as ordinary and as linear homogeneous, linear nonhomogeneous or nonlinear differential equations, state the order of the equations and name the dependent and
Classify the following differential equation problems as under-determined, fully determined or over-determined, and solve them where possible: (a) =t, x(0) = 1 dx dt (b) dx dt (d) d.x (c) = sin t,
Sketch the direction field of the differential equationand sketch the form of solution suggested by the direction field. Solve the equation and confirm that the solution supports the inferences you
Solve the following differential equation problems: dx cost (a) + dt sin x d.x (b) te=0, x(1)=2 dt = 0, Xx(0) = -
For each of the following problems, determine which are exact differentials, and hence solve the differential equations where possible: (a) 2x1 dx = a - 2x4, x(1) = 2 dt (b) (2xt + 2t + f) dx + x +
Solve the following differential equation problems: (a) d.x dt dx (b) + 2tx=(t)e, x(0) = 1 dt (d) -2x = 1, x(0) = 2 dx (c) + 3x = e, x(0) = 2 dt d.x dt + x sint sint, x(T) = e
Solve the differential equation to find the value of X(0.4) using Euler’s method with step size 0.1 and 0.05. By comparing these two estimates of x(0.4), estimate the accuracy of the better of the
Solve the differential equationto find the value of X(0.25) using Euler’s method with steps of size 0.05 and 0.025. By comparing these two estimates of x(0.25), estimate the accuracy of the better
Solve the differential equation obtained in Example 8.4 to determine the amount x(t) of salt in the tank at time t minutes. Initially the tank contains pure water.Data from Example 8.4Suppose that a
An open vessel is in the shape of a right-circular cone of semi-vertical angle 45° with axis vertical and apex downwards. At time t = 0 the vessel is empty. Water is pumped in at a constant rate
Stefan’s law states that the rate of change of temperature of a body due to radiation of heat iswhere T is the temperature of the body, T0 is the temperature of the surrounding medium (both
A motor under load generates heat internally at a constant rate H and radiates heat, in accordance with Newton’s law of cooling, at a rate kθ, where k is a constant and θ is the temperature
A linear cam is to be made whose rate of rise (as it moves in the negative x direction) at the point (x, y) on the profile is equal to one-half of the gradient of the line joining (x, y) to a fixed
Radioactive elements decay at a constant rate per unit mass of the element. Show that such decays obey equations of the formwhere k is the decay rate of the element and m is the mass of the element
In Section 10.2.4 we showed that the equation governing the current flowing in a series LRC electrical circuit is (equation (10.9))Show, by a similar method, that the equation governing the current
The tread of a car tyre wears more rapidly as it becomes thinner. The tread-wear rate, measured in mm per 10 000 miles, may be modelled as a + b(d – t)2 where d is the initial tread depth, t is
Express each of the following differential equations in the form (a) L[x(1)] = f(t) dx + (sin t) - 9x + cost = 0 dt dx dt
For each of the following pairs of operators calculate the operator LM – ML; hence state which of the pairs are commutative (that is, satisfy LMx(t) = MLx(t)): (a) L= (b) L= (c) L = (d) L = d dt d
What conditions must the functions f(t) and g(t) satisfy in order for the following operator pairs to be commutative? (a) L = (b) L= d dt d dt + f(t), M=. dt + f(t), M= + g(t) d = + g(t) dr
Solve the following initial-value problems: (a) (c) dx dt (d) dx d.x (b) 3-2-x = 2t - 1, dr dt x(0) = 7, x(0) = 0, dx dt d.x 2- + 5x = 1, x(0) = 0, dt dx d.x +2- dt dt dx dt - - d.x dx dt + dt d.x dt
Show that by making the substitutionShow that the solution of this equation is v = 1 + Ce–t and hence find x(t). This technique is a standard method for solving second-order differential equations
Using the method introduced in Question 24, find the solutions of the following initial-value problems:Data from Question 24Show that by making the substitutionShow that the solution of this equation
Show that by making the substitutionShow that the solution of this equation is v = 1/2x2 + C and hence find x(t). This technique is a standard method for solving second-order differential equations
Using the method introduced in Question 26, find the solutions of the following initial-value problems:Data from Question 26Show that by making the substitutionShow that the solution of this equation
Equation (10.3), arising from the model of the take-off run of an aircraft developed, can be solved by the techniques introduced in Exercises 24 and 26. Assuming that the thrust is constant, find the
Find the values of X(t) for t up to 2, where X(t) is the solution of the differential equation problemusing Euler’s method with step size h = 0.025. Repeat the computation with h = 0.0125. Hence
The end of a chain, coiled near the edge of a horizontal surface, falls over the edge. If the friction between the chain and the horizontal surface is negligible and the chain is inextensible then,
A simple mass–spring system, subject to light damping, is vibrating under the action of a periodic force Fcos pt. The equation of motion iswhere F and p are constants. Solve the differential
An alternating emf of Esinωt volts is supplied to a circuit containing an inductor of L henrys, a resistor of R ohms and a capacitor of C farads in series. The differential equation satisfied by the
A truck of mass m moves along a horizontal test track subject only to a force resisting motion that is proportional to its speed. At time t = 0 the truck passes a reference point moving with speed U.
Figure 10.37 shows a system that serves as a simplified model of the phenomenon of ‘tool chatter’. The mass A rests on a moving belt and is connected to a rigid support by a spring. The
The second-order, linear, nonhomogeneous constant-coefficient differential equation(often referred to as a forced harmonic oscillator) has a responseis often called the frequency response (strictly
Classify the equations (10.3), (10.6), (10.7) and (10.9) derived in the engineering examples of Section 10.2.(a)(b)(c)(d) ds dt m- - - ( ) ds dt = Tumg (10.3)
Find the general solution of the differential equation dx dt = t - 3et
Find the function x(t) that satisfies the differential equationand that has the value 2.5 when t = 0. dx dt = -4x ||
Find the function x(t) that satisfies the initial-value problem dx dt +2x = 0 x(0) = 4, d.x - (0) = 3, = 0 dt
Find the function x(t) that satisfies the boundary-value problem dx dt +2x = 0 x(0) = 4, dx ( dt = 3, A # 0
Sketch the direction field of the differential equationVerify that x(t) = Ce–t/2 is the general solution of the differential equation. Find the particular solution that satisfies x(0) = 2 and
Solve the equation dx/dt = 4xt, x > 0
Solve the equation 2 dx = x + xt, t > 0, x #0 dt
Solve the first-order linear differential equation dx dt + tx = t
Solve the first-order linear initial-value problem dx 1 += x=t, x(2) = dt t
The function x(t) satisfies the differential equationand the initial condition x(1) = 2. Use Euler’s method to obtain an approximation to the value of x(2) using a step size of h = 0.1. dx dt x + t
Let Xa denote the approximation to the solution of the initial-value problemobtained using Euler’s method with a step size h = 0.1, and Xb that obtained using a step size of h = 0.05. Compute the
Compute an approximation X(1) to the value of x(1) satisfying the initial-value problemby using Euler’s method with a step size h = 0.008. dx dt x t +1' x(0) = 1
The equationis a linear differential equation. Identify the operator of the equation and show that (10.36) holds for this operator. dx dt d.x +4t- dt (sint) x = cost
Find the general solution of the equation d4x dr4 - 14x = 0
Find the general solution of the differential equation dx dt dx 2- dt dx dt + 2x = 0
Which of the following sets of functions are linearly dependent and which are linearly independent?(a) {1 + t, t, 1}(b) {1 + t, 1 + t + t2, 1 + t2}(c) {sin(t), cos(t)}(d) {et, e2t, e3t}
Find the general solution of the differential equation dx dt + 2x=41, X>0
Find the general solution of the boundary-value problem dx dt IC - kx = sin 2t, k> 0, x(0) = 0, x 4 = 0
Find the general solution of the equation dx dt d.x 0=X9 +6 dt
Find the general solution of the equation dx 2- dt dx 3+5x = 0 dt
Find the general solution of the equation dx dt3 dx dt dx - 5- + 6x = 0 dt
Find the general solution of the equation d4x dx +3- dt4 dt dx 22- dt dx -73- dt - - 60x = 0
Find the general solution of the equation d4x dx dt4 + dt3 dx dt 3- dx 5- dt 2x = 0
Find the general solution of the equation dx dx +5- - 9x = 1 dt dt
Find the general solution of the equation dx dz dx + 5 - dt 9x = 9x = cos 2t
Find the general solution of the equation dx dt +5dx _ 9x = e4r dt
Find the general solution of the equation dx dx +59x = e-2 +2 - t dt dt
Find the general solution of the equation dx dx dt dt 2x = e-2t
Find the general solution of the equation d4x dt4 - dx 2- dt3 5dx dt +5- - dx 8 + 4x = e dt
A pendulum of mass 4 kg, length 2 m and an air resistance coefficient of 5 Nsm–1 is released from an initial position in which it makes an angle of 20° with the vertical. Assuming that this angle
Find the value of X(1.4) satisfying the following initial-value problem:using Euler’s method with time step h = 0.1. dx dt dy dt = x y + xt, = 2x + xy - t, x(1) = 0.5 y(1) = 1.2
Find the value of X(0.2) satisfying the initial-value problemusing Euler’s method with step size h = 0.05. dx dx dx dt dt +xt- +1 -1x = 0, x(0) = 1, dt dx -(0) = dt 0.5, dx dt (0) = = -0.2
Draw a phase-plane direction field for the equationHence sketch the solution path of the equation that starts from the initial conditions x = 1, dx/dt = 0. dx dt +1.5 d.x dt + 40x = 0 (10.60)
1. In a five-match test series, the runs scored by the two opening batsmen were as given in Table 10.3. Calculate the mean score over the series for each batsman.2. The grouped data of Table 10.4
1. Determine the median of the set of percentage marks obtained in a mathematics examination and presented in Table 10.1.2. Find the median scores of the two opening batsmen recorded in Table 10.3.3.
1. A small organisation employs 13 staff and their weekly salaries are given in Table 10.11. Determine the mode of the 13 salaries.2. Determine the mode of the plant height data of Table 10.5.3. A
1. Construct a cumulative frequency table of the plant height data in Table 10.5 and draw a cumulative graph. Determine the lower quartile, upper quartile, median and inter-quartile range of the

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