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mathematics
advanced engineering mathematics
Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions
Solve the given ODE. Show the details of your work.y"' + 4y" + 13y' = 0
Are the given functions linearly independent or dependent on the half-axis x > 0? Give reason.cosh 2x, sinh 2x, e2x
Find out, without calculation, whether doubling the flow rate in Example 1 has the same effect as halfing the tank sizes. (Give a reason.)
Find a general solution by conversion to a single ODE.The system in Example 5 of the text.
Determine the radius of convergence. Show the details of your work. m-0 (-1) km 2m x
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.y" + (x - 1)y = 0
List some properties of the Legendre polynomials.
Graph Q0(x), Q1(x), and some further Legendre functions.
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated substitutions. Show the details of your work.(2x + 1)2y" + 2(2x + 1)y' + 16x(x + 1)y = 0 (2x + 1 = z)
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.xy" + y' - xy = 0
Why did we introduce two kinds of Bessel functions?
Substitute αsxs + αs+1xs+1 + αs+2xs+2 into Legendre’s equation and obtain the coefficient recursion (4).
Apply the power series method. Do this by hand, not by a CAS, to get a feel for the method, e.g., why a series may terminate, or has even powers only, etc. Show the details.y" + y = 0
Find a general solution in terms of Jv and Yv. Indicate whether you could also use J-v instead of Yv. Use the indicated substitution. Show the details of your work.xy" - 5y' + xy = 0 (y = x3u)
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated substitutions. Show the details of your work.xy" + (2v + 1)y' + xy = 0 (y = x-v u)
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.2x(x - 1)y" - (x + 1)y' + y = 0
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated substitutions. Show the details of your work.x2y" + (1 - 2v )xy' + v2(x2v + 1 - v2)y = 0 (y = xvu, xv = z)
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.y" + 4y = 0
Find a power series solution in powers of x. Show the details.y" - y' + x2y = 0
Find a solution of (α2 - x2)y" - 2xy' + n(n+1)y = 0, α ≠ 0, by reduction to the Legendre equation.
Show that the Hankel functions (10) form a basis of solutions of Bessel’s equation for any v.
Find and graph (on common axes) the solutions of for k = 0, 1, 2, . . . ., 10 (or as far as you get useful graphs). For what k do you get elementary functions? Why? Try for noninteger k, particularly between 0 and 2, to see the continuous change of the curve. Describe the change of the location of
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.xy" + (2 - 2x)y' + (x - 2)y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.(x - 1)2y" - (x - 1) y' - 35y = 0
Show that Iv(x) has the representation 1,(x) = Σ m-0 2m + v X 22m+ +vm! F(m + v + 1)
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.xy" + (1 - 2x)y' + (x - 1)y = 0
Find a power series solution in powers of x. Show the details.y" - 4xy' + (4x2 - 2)y = 0
Find a general solution in terms of hyper geometric functions.2x(1 - x)y" - (1 - 6x)y' - 2y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.x2y" + xy' + (x2 - 5)y = 0
Shifting summation indices is often convenient or necessary in the power series method. Shift the index so that the power under the summation sign is Xm. Check by writing the first few terms explicity. s(s+ 1) 2 s² + 1 8-2 -x³-1, -XP+4 P-1(P + 1)! Р
Associated Legendre functions Pkn(x)?are needed, e.g., in quantum physics. They are defined by and are solutions of the ODE where q(x) = n(n + 1) - k2/(1 - x2). Find P11(x), P12(x), P22(x), and P24(x) and verify that they satisfy (16). P(x) = (1 - x²) x/2 d Pn(x) – dxk (1-x²)y" - 2xy + q(x)y
Modified Bessel functions of the third kind (Sometimes called of the second kind) are defined by the formula (14) below. Show that they satisfy the ODE (12). K₂(x) = TT 2 sin VTT - [I_₂(x) — 1₂(x)].
ZEROS of Bessel functions play a key role in modelingUsing (21) and Rolle’s theorem, show that between any two consecutive zeros of J0(x) there is precisely one zero of J1(x).
Find a power series solution in powers of x. Show the details.y" + (1 + x2)y = 0
Find a general solution in terms of hyper geometric functions.x(1 - x)y" + (1/2 + 2x)y' - 2y = 0
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.y' + 4y = 1, y(0)= 1.25, x1 = 0.2
Show that y = uv with v(x) = exp (-1/2 ∫ p(x) dx) gives from the ODE y" + p(x)y' + q(x)y = 0 the ODEu" + [q(x) - 1/4p(x)2 - 1/2p' (x)] u = 0,not containing the first derivative of u.
Find a general solution in terms of hyper geometric functions.4x(1 - x)y" + y' + 8y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.xy" - (x + 1)y' + y = 0
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.y" + 3xy' + 2y = 0, y(0)= 1, y' (0) = 1, x = 0.5
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.y" + 3xy' + 2y = 0, y(0)= 1, y' (0) = 1, x = 0.5
Find a general solution in terms of hyper geometric functions.2(t2 - 5t + 6)ÿ - (2t - 3)ý - 8y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.y" + (1/4x)y = 0
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.(x - 2)y' = xy, y(0) = 4, x1 = 2
Find a general solution in terms of hyper geometric functions.3t(1 + t)ÿ + tý - y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.xy" + y' - xy = 0
Linear Independence is of basic importance, in this chapter, in connection with general solutions, as explained in the text. Are the following functions linearly independent on the given interval? Show the details of your work.x2, x2 In x, x > 1
Linear Independence is of basic importance, in this chapter, in connection with general solutions, as explained in the text. Are the following functions linearly independent on the given interval? Show the details of your work.sin 2x, cos x sin x, x < 0
If the mathematics scores of the SAT college entrance exams are normal with mean 480 and standard deviation 100 (these are about the actual values over the past years) and if some college sets 500 as the minimum score for new students, what percent of students would not reach that score?
State the main theorems on probability. Illustrate them by simple examples.
Determine the number of different bridge hands. (A bridge hand consists of 13 cards selected from a full deck of 52 cards.)
Let X be the number of years before a certain kind of pump needs replacement. Let X have the probability function f(x) = kx3, x = 0, 1, 2, ,3, 4. Find k. Sketch f and F.
Graph a sample space for the experiments:Tossing a coin until the first Head appears
Graph a sample space for the experiments:Rolling 2 dice
Graph f and F when the density of X is f(x) = k = const if -2 ≤ x ≤ 2 and 0 elsewhere. Find P(0 ≤ X ≤ 2).
Let f(x, y) = k if x > 0, y > 0, x + y < 3 and 0 otherwise. Find k. Sketch f(x, y). Find P(X + Y ≤ 1), P(Y > X).
Find the adjacency matrix of the given graph or digraph. (1) еб е1 e A ег (3) (2) ез
State the most important facts about distributions of two random variables and their marginal distributions.
Graph a sample space for the experiments:Drawing gaskets from a lot of 10, containing one defective D, unitil D is drawn, one at a time and assuming sampling without replacement, that is, gaskets drawn are not returned to the lot.
Let X [millimeters] be the thickness of washers. Assume that X has the density f(x) = kx if 0.9 < x < 1.1 and 0 otherwise. Find k. What is the probability that a washer will have thickness between 0.95 mm and 1.05 mm?
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Efficiency [%] of seven Voith Francis turbines of runner diameter 2.3 m under a head range of 185 m91.8 89.1 89.9 92.5 90.7 91.2 91.0
In 1910, E. Rutherford and H. Geiger showed experimentally that the number of alpha particles emitted per second in a radioactive process is a random variable X having a Poisson distribution. If X has mean 0.5, what is the probability of observing two or more particles during any given second?
If the diameter X [cm] of certain bolts has the density f(x) = k(x - 0.9)(1.1 - x) for 0.9 < x < 1.1 and 0 for other x, what are k, μ, and σ2? Sketch f(x).
In how many different ways can 6 people be seated at a round table?
If we inspect photocopy paper by randomly drawing 5 sheets without replacement from every pack of 500, what is the probability of getting 5 clean sheets although 0.4% of the sheets contain spots?
A manufacturer knows from experience that the resistance of resistors he produces is normal with mean μ = 150 Ω and standard deviation σ = 5 Ω. What percentage of the resistors will have resistance between 148 Ω and 152 Ω? Between 140 Ω and 160 Ω?
Graph a sample space for the experiments:Recording the daily maximum temperature X and the daily maximum air pressure Y at Times Square in New York
Let X be the number of cars per minute passing a certain point of some road between 8 A.M. and 10 A.M. on a Sunday. Assume that X has a Poisson distribution with mean 5. Find the probability of observing 4 or fewer cars during any given minute.
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Release time [sec] of a relay1.3 1.2 1.4 1.5 1.3 1.3 1.4 1.1 1.5 1.41.6 1.3 1.5
When is the Poisson distribution a good approximation of the binomial distribution? The normal distribution?
What are the mean thickness and the standard deviation of transformer cores each consisting of 50 layers of sheet metal and 49 insulating paper layers if the metal sheets have mean thickness 0.5 mm each with a standard deviation of 0.05 mm and the paper layers have mean 0.05 mm each with a standard
Of a lot of 10 items, 2 are defective.(a) Find the number of different samples of 4. Find the number of samples of 4 containing(b) No defectives(c) 1 defective(d) 2 defectives.
What is sampling with and without replacement? What distributions are involved?
If the lifetime X of a certain kind of automobile battery is normally distributed with a mean of 5 years and a standard deviation of 1 year, and the manufacturer wishes to guarantee the battery for 4 years, what percentage of the batteries will he have to replace under the guarantee?
Five fair coins are tossed simultaneously. Find the probability function of the random variable X = Number of heads and compute the probabilities of obtaining no heads, precisely 1 head, at least 1 head, not more than 4 heads.
Graph f and F when f(-2) = f(2) = 1/8, f(-1) = f(1) = 3/8. Can f have further positive values?
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Weight of filled bags [g] in an automatic filling203 199 198 201 200 201 201
State the definition of probability from memory. Give simple examples.
Find the mean and variance of the random variable X with probability function or density f(x).f(x) = 4e4x (x ≥ 0)
Find the density of the marginal distribution of Y in Fig. 524. az 0 01 B,
In how many different ways can we select a committee consisting of 3 engineers, 2 physicists, and 2 computer scientists from 10 engineers, 5 physicists, and 6 computer scientists? First guess.
If a box contains 10 left-handed and 20 right-handed screws, what is the probability of obtaining at least one right-handed screw in drawing 2 screws with replacement?
What is a random variable? Its distribution function? Its probability function or density?
Let X be normal with mean 50 and variance 9. Determine c such that P(X < c) = 5%, P(X > c) = 1%, P(50 - c X < 50 + c) = 50%
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Systolic blood pressure of 15 female patients of ages 20–22156 158 154 133 141 130 144 137151 146 156
Find the mean and variance of the random variable X with probability function or density f(x).Uniform distribution on [0, 2π]
What do we mean by an experiment? An outcome? An event? Give examples.
If a box contains 4 rubber gaskets and 2 plastic gaskets, what is the probability of drawing(a) First the plastic and then the rubber gaskets(b) First the rubber and then the plastic ones? Do this by using a theorem and checking it by multiplying probabilities.
Three screws are drawn at random from a lot of 100 screws, 10 of which are defective. Find the probability of the event that all 3 screws drawn are nondefective, assuming that we draw(a) With replacement(b) Without replacement.
What properties of data are measured by the mean? The median? The standard deviation? The variance?
In rolling 2 fair dice, what is the probability of a sum greater than 3 but not exceeding 6?
Let X be normal with mean 10 and variance 4. Find P(X > 12), P(X < 10), P(X < 11), P(9 < X < 13).
Graph a sample space for the experiments:Drawing 3 screws from a lot of right-handed and left handed screws
Graph the probability function f(x) = kx2 = (x = 1, 2, 3, 4, 5; k suitable) and the distribution function.
Mark the positions of ? in Fig. 517. Comment. 0.5 p = 0.1 p = 0.2 p = 0.5 GL 5 p = 0.8 0 p = 0.9
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Length of nails [mm]19 21 19 20 19 20 21 20
What are stem-and-leaf plots? Boxplots? Histograms? Compare their advantages.
Find the mean and variance of the random variable X with probability function or density f(x).f(x) = kx (0 ≤ x ≤ 2, k suitable)
Let f(x, y) = k when 8 ≤ x ≤ 12 and 0 ≤ y ≤ 2 and zero elsewhere. Find k. Find P(X ≤ 11, 1 ≤ Y ≤ 1.5) and P(9 ≤ X ≤ 13, Y ≤ 1).
In how many ways can a company assign 10 rivers to n buses, one driver to each bus and conversely?
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