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modern engineering mathematics

Modern Engineering Mathematics 6th Edition Glyn James - Solutions

Denote Euler’s method solution of the initial-value problemusing step size h = 0.05 by Xa(t), and that using h = 0.025 by Xb(t). Find the values of Xa(1.5) and Xb(1.5). Estimate the error in the value of Xb(1.5), and suggest a value of step size that would provide a value of X(1.5) accurate to
For each of the following differential equations¥write down the differential operator¥L¥that would enable the equation to be expressed¥as L[x(t)] = 0: d.x (a) + 1x=0 dt dx (c) - kx = 0 dt dx dt (b)- = 6xt
Which of the following two sets are linearly dependent and which are linearly independent? (a) {1, 1, 1, 1, 14, 15, 1} (b) {1+t, t, t t, 1 t} - -
For each of the following sets of linearly dependent functions find k1, k2, . . . . such that k1f1 + k2f2 + . . . . = 0. (a) {t + 1, 1,2} (b) {t - 1,t + 1,1 - 1, t + 1}
For each of the following differential equations write down the differential operator L that would enable the equation to be expressed as L[x(t)] = 0: dx (a) = f(t)x dt (b) (c) (e) 60 dx dt (h) dx dt (d) (sin t)- + (sin t)- dx d.x + (sin t) = (t + cost)x dt dt d.x bx dt t d ,d.x 1. dt dt dx dt dt t
Which of the following sets of functions are linearly dependent and which are linearly independent? (a) (sint + 2 cost, sint - 2 cost, 2 sint + cost, 2 sint - cost} (b) {sint, cost, sin 2t, cos 2t, sin 3t, cos 3t} (c) {1+2t, 2t - 31, 31 + 4t, 4t - 5t4} (d) {1+ 2t, 2t - 3t, 3t + 4t, 41} (e) {1, 1+
For each of the following sets of linearly dependent functions find k1, k2, . . . . such that k1f1 + k2 f2 + . . . . = 0: (a) (sint, cost + sint, cos 2t - sint, cost - cos 2t} (b) {t+t,t-t, t + 21, 1 - 1} (c) (Int, In 2t, In 4t} (d) {f(t) + g(t), f(t)(1 + f(t)), g(t) - f(t), f(t) - g(t)} (e) {1+t+
Determine which members of the given sets are¥solutions of the following differential equations. Hence, in each case, write down the¥general solution of the differential equation. (a) dx dt4 = 0 {1, t, t, 1, t, ts, t} dx (b) --px = 0 {e, e, cos pt, sinpt} dt
The operators L and M are defined byFind L[M[x(t)]]. Hence write down the operator LM. Find M[L[x(t)]]. Is LM = ML? and L = d dt 1 d M = - t - dt 4t- + 6t dt
The operators L and M are defined byFind expressions for the operators LM and ML. Under what conditions on f1, g1, f2 and g2 is LM = ML? What conditions do you think linear differential operators must satisfy in order to be commutative? and L = f(0) + dt + g, (t) M = f(1) + 8(1)
Solve the following initial-value problems: (a) 5 dx dr (b) (c) dx dt2 dx dt d.x dt dx -2x = 0, x(0) = -1, (0) = 1 dt dx d.x 6+10x=0, x(0) = 2,(0) = 0 dt dt dx dx 4- +3x = 0, x(0) = 0,= (0) = 1 dt dt
Find the general solutions of the following differential equations: 4dx (a) dt (b) (d) (e) dx dz (8) dx dr 2dx + 7x=0 dt dx dx (c) 3- +3- + 3x = 0 dt dt dx (f) dt dx -9 + dt d.x 8- dt - 4x = 0 9dx_9dx dt dt dx dt + 16x=0 dx dx dt dt d.x 4- + 4x = 0 dt 7.d.x dt + 9x = 0 dx +3=0 dt
Show that the characteristic equation of the differential equationand hence find the general solution of the equation. dx dx 4- +11- dt* dt dx dt d.x 14+ 10x = 0 dt
Solve the following initial-value problems: (a) 2- (b) (c) dx dt (e) dx dt dx dz (d) 9- dx dr dx dt + d.x - 2- dx dr x(0)= 1, dt dx dt 5 d.x dt d.x + 4x=0, x(1) = 0,(1)=2 dt d.x dx +6+x=0, x(-3) = 2,(-3) = // dt dt + 6- d.x + 3x = 0, x(0) = 1, (0) = 0 dt d.x + 8x = 0, x(0) = 1, (0) = -2 dt dx dt dx
Show that the characteristic equation of the differential equationand hence find the general solution of the equation. is d'x dt4 dx 2- + dt dt (mm + 1) = 0 dx dt + x = 0
Show that the characteristic equation of the differential equationand hence find the general solution of the equation. is dx__dx_9dx dt dt dt 11dx dt - 4x = 0 (m +3m +3m + 1)(m 4) = 0 -
Find the general solutions of the following differential equations: (a) (b) (d) dx dt (i) dx dz d.x dx (c) 2- +4- dt dt dx (e) 16- dt (j) (k) (h) 3dx dt dx dx + dt dt dx dt dx dt dx dt dx dt dx dr dx dt dx 12- + 4x = e-r dt + 4x = cos 4t - 2 sin 4t d.x + 8 + x = t + 6 dt + 4x = 5t - 7 dx 8 + 16x =
Show that the characteristic equation of the differential equationand hence find the general solutions of the equations is d+x dt dx 3- dx +9-2x = 0 dt dt dt dx 5- (m + m - 2)(m - 4m + 1) = 0
Show that the characteristic equation of the differential equationand hence find the general solutions of the equations is dx dt dx 9- df +27x - 27x=0 dt (m3) = 0
Find the damping parameters and natural frequencies of the systems governed by the following second-order linear constant-coefficient differential equations: (a) (b) dx dt dx dt dx + 6- + 9x=0 dt d.x +4- + 7x=0 dt
Determine the values of the appropriate parameters needed to give the systems governed by the following second-order linear constant-coefficient differential equations the damping parameters and natural frequencies stated: (a) (b) (c) dx dt dx dr dx dt d.x + 2a + bx = 0, = 0.5, @= 2 dt dx + P dt
Find the damping parameters and natural frequencies of the systems governed by the following second-order linear constant-coefficient differential equations: (a) dx dt (b) 2- (d) dx dt d.x + 2a- dt dx dt (c) 2.41- + 14 dx dt 1 dx 17 dt (e) 1.88- +16px = 0 1 +x=0 1.88dx dt +1.02dx +1.02 + 7.63x =
Determine the values of the appropriate parameters needed to give the systems governed by the following second-order linear constant-coefficient differential equations the damping parameters and natural frequencies stated: (a) (b) dx dt dx dr + (d) a- dx + a- dt d.x dt dx (c) 4- +9- dt + bx = 0,
The function A(Ω) is as given by (10.55) and shown in Figure 10.27. Show that A(Ω) has a simple maximum point when ζ max. Find Ωmax as a function of ζ and ω , and also find A(Ωmax). For >, A(2) has no maximum, but does have a single point of inflection. Show, by consideration of Figure
An underwater sensor is mounted below the keel of the fast patrol boat shown in Figure 10.29. The supporting bracket is of cylindrical cross-section (diameter 0.04 m), and so is subject to an oscillating side-force due to vortex shedding. The bracket is of negligible mass compared with the sensor
The piece of machinery shown in Figure 10.30 is mounted on a solid foundation in such a way that the mounting may be characterized as a rigid pivot and two stiff springs as shown. A damper is connected between the machine and an adjacent strong point. The mass of the machine is 500 kg, the length a
Figure 10.31 shows a radio tuner circuit. Show that the natural frequency and damping parameters of the circuit are 1/√(LC) andrespectively. If R1 = 300Ω and R2 = 50Ω what value should L have, and over what range should C be adjustable in order that the circuit have a damping factor of ζ = 0.1
Transform the following initial-value problems into sets of first-order differential equations with appropriate initial conditions: dx (a) + 6(x-t) - 4xt = 0, dr d.x dt (b) dx x(0) = 1, d (0)= 2 dt dx dt sin x(0) = 0, d.x dt + 4x = 0, d.x -(0) = 0 dt
Find the value of X(0.3) for the initial-value problemusing Euler’s method with step size h = 0.1. dx dt 2 dx + x = sint, dt +x. x(0) = 0, d.x -(0) = 1 dt
Transform the following initial-value problems into sets of first-order differential equations with appropriate initial conditions: (a) (b) (c) (d) (f) dr dx x(1)=2, (1) = 0.5 dx dx +1- dr dr dx dr + 4(x - 1)/ = 0, dx dr x(1) = 1, d.x x(0) = 1, (0) = 2, dx dr dx dr + x(0) dx dt 1/2 -x+ -(0) = 0, -
Use Euler’s method to compute an approximation X(0.65) to the solution x(0.65) of the initial-value problemusing a step size of h = 0.05. dx dx + dt dt d.x dt (0.5)=1, (0.5)=2 -(x-t) + x(0.5) = -1, d.x dt - x = 0,
Write a computer program to solve the initialvalue problemusing Euler’s method. Use your program to find the value of X(0.4) using steps of h = 0.01 and h = 0.005. Hence estimate the accuracy of your value of X(0.4) and estimate the step size that would be necessary to obtain a value of X(0.4)
A water treatment plant deals with a constant influx Q of polluted water with pollutant concentration s0. The treatment tank contains bacteria which consume the pollutant and protozoa which feed on the bacteria, thus keeping the bacteria from increasing too rapidly and overwhelming the system. If
Draw phase-plane direction fields for the following equations and sketch the form you would expect the solution paths to take, starting from the points (x, v) = (1, 0), (0, 1), (–1, 0) and (0, –1) in each case: dx dx (a) + - + x = 0 dt dt dx d.x (b) + + sgn(x) = 0 dr dt
For each of the problems in Question 78 solve the differential equation numerically and check that the solutions you obtain are similar to your sketch solutions.Data from Question 78Write a computer program to solve the initialvalue problemusing Euler’s method. Use your program to find the value
In an ice-skating competition, the total scores awarded by the judges to each of the seven competitors were as given in Table 10.6.(a) Determine the median score.(b) Why is the median score a realistic choice for determining the ‘typical score’? Competitor X1 X2 X3 X4 X5 X6 X7 Score 112 174 165
Determine the mode of the set of examination marks given in Table 10.1. Population (students) X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 Variable/Observation (% marks) 60 55 60 48 70 38 52 54 66 42 Population Variable/Observation (students) (% marks) X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 Table 10.1 % marks
Determine the (a) Range, (b) Upper quartile, (c) Lower quartile(d) Inter-quartile range, of the data set 6, 7, 11, 12, 15, 17, 19, 20, 23, 24, 26, 28, 29, 30, 32
Determine the mean deviation of the data set given in Table 10.1, which represents the marks obtained in a mathematics examination by a group of 20 students. Population (students) X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 Variable/Observation (% marks) 60 55 60 48 70 38 52 54 66 42 Population
Calculate the standard deviation of the data set given in Table 10.1, which represent the marks obtained in a mathematics examination by a group of 20 students. Population (students) X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 Variable/Observation (% marks) 60 55 60 48 70 38 52 54 66 42 Population
Evaluate the following, if they are defined: fronda 0 (a) 23 dx (b) dx (1-x) (c) [T 0 Inx dx (d) S. dx
Confirm that is not defined. f(1/x)dx
Obtain the value of the integralwhere α = π/3. This is the period of oscillation of a simple pendulum released from rest from the angle a. D ra de >- (), V(sin 4 - sin 4) 2 0 T() = 2
Evaluate the following: (a) Si 00 x-3/2 dx (b) dx 1 + x (c) 0 et sinx dx (d) -00 e exp(-e)dx
Show thatand hence estimate an error bound for sin a, where a = 1.935 (3dp). Compare the error interval obtained with [sin 1.9355, sin 1.9345]. Express sin a as a correctly rounded number with the maximum number of decimal places. A(sin x) cos.x Ax
Show that the iterationis convergent to the root near θ = 3.9 of the equation tan θ = tanh θ (see Figure 9.4). On+ tan-'(tanh0,), with 0o = t ~ 3.9
A spherical wooden ball floats in water, as illustrated in Figure 9.5. Its diameter is 10 cm and its density is 0.8 gcm−3. Find the depth h cm to which it sinks. Figure 9.5 Floating ball 10 cm h cm
Find a polynomial approximation to the function f(x) such that f(0)= 3, f'(0) = 4, f"(0) = -10 and f""(0) = 12
Find a polynomial approximation to f(x) such that f(1) = 4, f'(1) = 0, f"(1) = 2 and f(1) = 12
Find the Maclaurin series expansion of exsin x.
Using the Maclaurin series expansions of ex and sin x, confirm the Maclaurin series expansion of exsin x obtained in Example 9.10.Data from Example 9.10Find the Maclaurin series expansion of exsin x.
Obtain the binomial expansion of (1 – x2)−1/2 and deduce a power series expansion for sin−1x.
The continuous belt of Example 1.48 has length L given byShow that when R – r ≪ l, a good approximation to L is given byData from Example 1.48A continuous belt of length L m passes over two wheels of radii r and R m with their centres a distance l m apart, as illustrated in Figure 1.32. The
Using L’Hôpital’s rule, obtain the limits sin x - x (a) lim- x->0 (b) lim 1 COS X x+0 x + x 2
The function f(x) = e−x is to be tabulated to 4dp on the interval [0, 0.5]. Find the maximum tabular interval such that the resulting table is suitable for linear interpolation to 4dp; that is, to yield an interpolated value which is as accurate as the tabulated value.
The equation x tan x = 4 has an infinite number of roots. To find the root near x = 1, we may use the iterationShow that this is a first-order process. Starting with x0 = 1, find x3 and assess its accuracy. Xn+1 tan = +| Xn
The equation x tan x = 4 was considered earlier in Example 9.16. Apply the Newton–Raphson method to find the root near x = 1.Data from Example 9.16The equation x tan x = 4 has an infinite number of roots. To find the root near x = 1, we may use the iterationShow that this is a first-order
Find the root ofnear x = 0.8 to 4sf. 8.0000x+0.4500x 4.5440x 0.1136 = 0 - -
In Example 8.72 the integralwas estimated using the trapezium rule with h = 1/4 and tabulating the integrand to 6dp. Estimate an error bound for the answer obtained.Data from Example 8.72Evaluate the integralusing the trapezium rule. xp(x/1)S
Sketch the curve r = sin t i + cos t j Calculate (a) dr dt (b) dr dt (c) dr dt d (d) (r) dt
Given find r(t). Obtain the locus of the point P, such that OP(vector) = r, in terms of x and z when V = (u, 0, v). dr dt = -gk with r(0) = 0 and r(0) = V
Find from first principlesand at the point (1, 2) where f(x, y) = x3 + 3xy + y2. af and af
Find from first principles the first partial derivatives of f(x, y) = y sin x at the general point (x, y).
Findwhere f(x, y) is given by(a) 3x2 + 2xy + y3 (b) (y2 + x)e−xy af X and af
Find ∂f/∂x and ∂f/∂y when f(x, y) is(a) xy2 + 3xy – x + 2 (b) sin(x2 – 3y)
Find the partial derivatives of f(x, y, z) = xyz2 + 3xy – z with respect to x, y and z.
Find the partial derivatives of f(x, y) = x2y3 + 3y + x with respect to x and y, and the slope of the function in the direction at an angle a to the x axis.
Find ∂T/∂r and ∂T/∂θ whenand x = r cosθ and y = r sin θ T(x, y) = x - xy + y
Find dH/dt when H(t) = sin(3x – y) and x = 2t - 3 and y=t5t + 1
The base radius r cm of a right-circular cone increases at 2 cms−1 and its height h cm at 3 cms−1. Find the rate of increase in its volume when r = 5 and h = 15.
Find dz/dt when (a) z = e cos y, where x = 2t + f and y = 4t (b) z = x + fand x +21 + 3xt = 0.
Find the second partial derivatives of f(x, y) = x2y3 + 3y + x.
Find the second partial derivatives of f(x, y, z) = xyz2 + 3xy – z
f(x, y) is a function of two variables x and y that we wish to change to variables s and t, where s = x2 – y2, t = xy. Determine fxx and fyy in terms of s, t, fs, ft, fss, ftt and fst. Show that fxx + fyy = (s + 4t)(4ss +f1)
Find the total differential of u(x, y) = x2y3.
The volume V cm3 of a circular cylinder of radius r cm and height h cm is given by V = πr2h. If r = 3 ± 0.01 and h = 5 ± 0.005 find the greatest possible error in the calculation of V and compare it with the estimate obtained using the total differential.
Two variables, x and y, are related by y = ae−bx, where a and b are constants. The values of a and b are determined from experimental data and have relative error bounds p and q respectively. What is the relative error bound for a value of y calculated using the formula with these values of a and
Show that (6x + 9y + 11)dx + (9x – 4y + 3)dy is an exact differential and find the relationship between y and x givenand the condition y = 1 when x = 0. dy dx 6x + y + 11 9.x - 4y + 3
Obtain the Taylor series of the function f(x, y) = sin xy about the point (1, 1/3π), neglecting terms of degree 3 and higher.
Find the stationary points of the function f(x, y) = 2x3 + 6xy2 – 3y3 – 150x and determine their nature.
Obtain the extremum value of the function f(x, y) = 2x2 + 3y2 subject to the constraint 2x + y = 1.
Rework Example 9.41 using the method of Lagrange multipliers.Data from Example 9.41Obtain the extremum value of the function f(x, y) = 2x2 + 3y2 subject to the constraint 2x + y = 1.
Find the dimensions of the cuboidal box, without a top, of maximum capacity whose surface area is 12 m2.
Apply the method of Lagrange multipliers to solve the design problem of Section 2.10.
Find the stream function Ψ(x, y) for the incompressible flow that is such that thevelocity q at the point (x, y) is (-y/(x + y), x/(x + y))
State the order of each of the following differential equations and name the dependent and independent variables. Classify each equation as linear homogeneous, linear nonhomogeneous or nonlinear differential equations. dx (a) + 2x = 0 dt dx (b) +2- dr (d) (e) dx dt d.x dt d.x dt + 3x = 0 + x = 0 +
Classify the following differential equations as linear homogeneous, linear nonhomogeneous or nonlinear differential equations, state their order and name the dependent and independent variables. (a) (b) (c) (d) (f) d'p dp dz dz (h) ds dr dp dy dr dz dx dt ds + (sin t) + (t + cost)s = e dt d.x (e)
Find the mean of the set of examination marks given in Table 10.1. Population (students) X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 Variable/Observation (% marks) 60 55 60 48 70 38 52 54 66 42 Population Variable/Observation (students) (% marks) X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 Table 10.1 % marks in
Determine the mean of the grouped data set of Table 9.7 (Unit 9) representing golf scores. Classes/intervals (score groups) 65-69 70-74 75-79 80-84 85-89 90-94 95-99 Tally HIT HIT 11 1 Total Frequency 1 7 11 12 5 3 1 40 Table 9.7 Tally/frequency table for grouped golf scores
Show that, if z = xeKxy, where K is a constant, then 0 = 2x pue 2 = 2 zx
The acceleration f of a piston is given byWhen θ = 1/6π radians and when r/L = 1/2, calculate the approximate percentage error in the calculated value of f if the values of both r and ω are 1% too small. f = roo(cost r(cos 0 + cos 26) 20 L
The area of a triangle ABC is calculated using the formula S = 1/2bc sin A and it is known that b, c and A are measured correctly to within 1%. If the angle A is measured as 45°, prove that the percentage error in the calculated value of S is not more than about 2.8%.
The angular deflection θ of a beam of electrons in a cathode-ray tube due to a magnetic field is given bywhere H is the intensity of the magnetic field, L is the length of the electron path, V is the accelerating voltage and K is a constant. If errors of up to ±0.2% are present in each of the
In a coal processing plant the flow V of slurry along a pipe is given byIf r and l both increase by 5%, and p and η decrease by 10% and 30% respectively, find the approximate percentage change in V. V= 8nl
Determine which of the following are exact differentials of a function, and find, where appropriate, the corresponding function. (a) (y + 2xy + 1)dx + (2xy +x) dy (b) (2xy + 3y cos 3x) dx + (2xy + sin 3x)dy (c) (6xy- y)dx + (2x - x) dy (d) (z - 3y)dx + (12y-3x)dy + 3xzdz
Find the value of the constant l such thatis the exact differential of a function f(x, y). Find the corresponding function f(x, y) that also satisfies the condition f(0, 1) = 0. (ycosx + Acos y)dx + (xsiny + sinx + y)dy $(ycosx + \ cos y)dx + (xsiny + sinx + y)dy$
Show that the differentialis not exact, but that a constant m can be chosen so thatis equal to dz, the exact differential of a function z = f(x, y). Find f(x, y). g(x, y) = (10x + 6xy + 6y)dx +(9x + 4xy + 15y) dy
Find the stationary values (and their classification) of (a) x 15x - 20y + 5 (b) 2x-xy - y (c) 2x + y + 3xy - 3y - 5x + 2 (d) x + y-3(x + y) + 1 (e) xy 2xy - 2x - 3x (f) xy(1-x-y) 2 2 (g) x + y + + x y -
Prove that (x + y)/(x2 + 2y2 + 6) has a maximum at (2, 1) and a minimum at (–2, –1).
Show thathas stationary values at (0, 0) and (1/3 , 1/3) and investigate their nature. f(x, y) = x + y - 2(x + y) + 3xy

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