Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 8.8, Problem 2P
By using L2, verify L7 and L8 in the Laplace transform table.
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Chapter 8 Solutions
Mathematical Methods in the Physical Sciences
Ch. 8.1 - Verify the statement of Example 2. Also verify...Ch. 8.1 - Solve Example 4 using the general solution...Ch. 8.1 - Verify that y=sinx,y=cosx,y=eix, and y=eix are all...Ch. 8.1 - Find the distance which an object moves in time t...Ch. 8.1 - Find the position x of a particle at time t if its...Ch. 8.1 - A substance evaporates at a rate proportional to...Ch. 8.1 - The momentum p of an electron at speed v near the...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...
Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - In Problems 13 to 15, find a solution (or...Ch. 8.2 - In Problems 13 to 15, find a solution (or...Ch. 8.2 - In Problems 13 to 15, find a solution (or...Ch. 8.2 - By separation of variables, find a solution of the...Ch. 8.2 - The speed of a particle on the x axis, x0, is...Ch. 8.2 - Let the rate of growth dN/dt of a colony of...Ch. 8.2 - (a) Consider a light beam traveling downward into...Ch. 8.2 - Consider the following special cases of the simple...Ch. 8.2 - Suppose the rate at which bacteria in a culture...Ch. 8.2 - Solve the equation for the rate of growth of...Ch. 8.2 - Heat is escaping at a constant rate [dQ/dtin(1.1)...Ch. 8.2 - Do Problem 23 for a spherical cavity containing a...Ch. 8.2 - Show that the thickness of the ice on a lake...Ch. 8.2 - An object of mass m falls from rest under gravity...Ch. 8.2 - According to Newtons law of cooling, the rate at...Ch. 8.2 - A glass of milk at 38 is removed from the...Ch. 8.2 - A solution containing 90 by volume of alcohol (in...Ch. 8.2 - If P dollars are left in the bank at interest I...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Water with a small salt content (5 lb in 1000 gal)...Ch. 8.3 - Find the general solution of (1.2) for an RL...Ch. 8.3 - Find the general solution of (1.3) for an RC...Ch. 8.3 - Prob. 18PCh. 8.3 - If 1=2= in (3.10), then e21tdt=dt. Find N2 for...Ch. 8.3 - Extend the radioactive decay problem (Example 2)...Ch. 8.3 - Generalize Problem 20 to any number of stages.Ch. 8.3 - Find the orthogonal trajectories of the family of...Ch. 8.3 - Find the orthogonal trajectories of the family of...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Solve the differential equation yy2+2xyy=0 by...Ch. 8.4 - If an incompressible fluid flows in a corner...Ch. 8.4 - Find the family of orthogonal trajectories of the...Ch. 8.4 - Find the family of curves satisfying the...Ch. 8.4 - Find the shape of a mirror which has the property...Ch. 8.4 - As in text just before (4.11), show that (a)...Ch. 8.4 - Show that the change of variables (4.13) in (4.11)...Ch. 8.4 - Show that (xP+yQ)1 is an integrating factor for...Ch. 8.4 - Solve Problems 9 and 10 by using an integrating...Ch. 8.4 - An equation of the form y=f(x)y2+g(x)y+h(x) is...Ch. 8.4 - Show that the substitution given in Problem 25...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Recall from Chapter 3, equation ( 8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation ( 8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation ( 8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation (8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation ( 8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation (8.5), that a set...Ch. 8.5 - Solve the algebraic equation D2+(1+2i)D+i1=0 (note...Ch. 8.5 - As in Problem 19, solve y+(1i)yiy=0. Hint: See...Ch. 8.5 - By the method used in solving (5.4) to get (5.9),...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Let D stand for d/dx, that is, Dy=dy/dx; then...Ch. 8.5 - In Example 3, we used the second solution in...Ch. 8.5 - A particle moves along the x axis subject to a...Ch. 8.5 - Find the equation of motion of a simple pendulum...Ch. 8.5 - The gravitational force on a particle of mass m...Ch. 8.5 - Find (in terms of L and C) the frequency of...Ch. 8.5 - A block of wood is floating in water; it is...Ch. 8.5 - Solve the RLC circuit equation [(5.33)or(5.34)]...Ch. 8.5 - (a) Find numerical values of the constants and...Ch. 8.5 - The natural period of an undamped system is 3 sec,...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Verify that (6.4) is a particular solution of...Ch. 8.6 - Solve (6.16) by the method used in solving (...Ch. 8.6 - Consider the differential equation...Ch. 8.6 - (a) Show that (Da)ecx=(ca)ecx;...Ch. 8.6 - (a) Show that Deaxy=eax(D+a)y, D2eaxy=eax(D+a)2y,...Ch. 8.6 - Using Problems 29 and 31b, show that equation...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - Find the solutions of (1.2) (put I=dq/dt ) and...Ch. 8.6 - In (6.38), show that for a given forcing frequency...Ch. 8.6 - Solve Problems 41 and 42 by use of Fourier series....Ch. 8.6 - Solve Problems 41 and 42 by use of Fourier series....Ch. 8.6 - Consider an equation for damped forced vibrations...Ch. 8.7 - Solve the following differential equations by...Ch. 8.7 - Solve the following differential equations by...Ch. 8.7 - Solve the following differential equations by...Ch. 8.7 - Solve the following differential equations by...Ch. 8.7 - The differential equation of a hanging chain...Ch. 8.7 - The curvature of a curve in the (x,y) plane is...Ch. 8.7 - Solve y+2y=0 by method (c) above and compare with...Ch. 8.7 - The force of gravitational attraction on a mass m...Ch. 8.7 - Show that (7.15) is a separable equation. [You may...Ch. 8.7 - In Problems 10 and 11, solve (7.14) to find v(x)...Ch. 8.7 - In Problems 10 and 11, solve (7.14) to find v(x)...Ch. 8.7 - In Problem 11, find v(x) if v=0,x=1, at t=0. Then...Ch. 8.7 - The exact equation of motion of a simple pendulum...Ch. 8.7 - Verify (7.19) and (7.20). Hint:...Ch. 8.7 - If you solve (7.17) when f(x)=0 by assuming a...Ch. 8.7 - Solve the following equations either by method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the two differential equations in Problem...Ch. 8.7 - Substitute (7.22) into (7.21) to obtain the...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.8 - For integral k, verify L5 and L6 in the Laplace...Ch. 8.8 - By using L2, verify L7 and L8 in the Laplace...Ch. 8.8 - Using either L2, or L3 and L4, verify L9 and L10.Ch. 8.8 - By differentiating the appropriate formula with...Ch. 8.8 - By integrating the appropriate formula with...Ch. 8.8 - By replacing a in L2 by a+ib and then by aib, and...Ch. 8.8 - Verify L15 to L18, by combining appropriate...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Show that a combination of entries L3 to L10, L13,...Ch. 8.8 - Prove L32 for n=1. Hint: Differentiate equation...Ch. 8.8 - Use L32 and L3 to obtain L11.Ch. 8.8 - Use L32 and L11 to obtain Lt2sinat.Ch. 8.8 - Use L31 to derive L21Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.9 - Continuing the method used in deriving (9.1) and...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.10 - Show that g*h=h*g as claimed in I34. Hint: Let u=t...Ch. 8.10 - Use L34 and L2 to find the inverse transform of...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the Laplace transform table to find...Ch. 8.10 - Use the convolution integral (see Example 2) to...Ch. 8.10 - Use the convolution integral (see Example 2) to...Ch. 8.10 - Consider solving an equation like (10.1) but with...Ch. 8.10 - Solve the differential equation ya2y=f(t), where...Ch. 8.10 - A mechanical or electrical system is described by...Ch. 8.10 - Following the method of equations (10.8) to...Ch. 8.11 - Find the inverse Laplace transform of e2p/p2 in...Ch. 8.11 - Verify L24 in the table by using L1, L27, and the...Ch. 8.11 - Verify L28 in the table by using L27 and the...Ch. 8.11 - Show that fn(t)dt=1 for the functions fn(t) in...Ch. 8.11 - Solve the differential equation y+2y=f(t),y0=y0=0,...Ch. 8.11 - (a) Let a mechanical or electrical system be...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Evaluate the functions fn(xa) defined by the...Ch. 8.11 - Using functions, write the following mass or...Ch. 8.11 - Integrate by parts as we did for (11.14) to obtain...Ch. 8.11 - Use (11.6) and (11.14) to (11.16) to evaluate the...Ch. 8.11 - Verify the operator equation ddxsgnx=2(x) where...Ch. 8.11 - Verify (11.18a) and (11.18c) by multiplying by a...Ch. 8.11 - Use equation (11.16) to generalize the operator...Ch. 8.11 - (a) Show that you can differentiate a generalized...Ch. 8.11 - Verify the operator equations in (11.19) not done...Ch. 8.11 - Make use of the operator equations (11.19) and...Ch. 8.11 - You may find the spherical coordinate function...Ch. 8.11 - Write a formula in rectangular coordinates, in...Ch. 8.11 - Prob. 24PCh. 8.11 - Let F(x)=x2,x0,0,x0. Show that F(x)=0 for all x0,...Ch. 8.12 - Solve (12.3) if G=0 and dG/dt=0 at t=0 to obtain...Ch. 8.12 - In Problems 2 and 3, use (12.6) to solve (12.1)...Ch. 8.12 - In Problems 2 and 3, use (12.6) to solve (12.1)...Ch. 8.12 - Use equation (12.6) to solve Problem 10.18.Ch. 8.12 - Obtain ( 12.6 ) by using the convolution integral...Ch. 8.12 - For Problem 10.17, show (as in Problem 1) that the...Ch. 8.12 - Use the Green function of Problem 6 to solve...Ch. 8.12 - Solve the differential equation...Ch. 8.12 - Following the proof of (12.4), show that (12.9)...Ch. 8.12 - Solve (12.12) and (12.14) to get (12.15). Hint:...Ch. 8.12 - In Problems 11 to 13, use (12.17) to find the...Ch. 8.12 - In Problems 11 to 13, use (12.17) to find the...Ch. 8.12 - In Problems 11 to 13, use (12.17) to find the...Ch. 8.12 - (a) Given that y1(x) and y2(x) are solutions of...Ch. 8.12 - In Problems 15 to 18, use the given solutions of...Ch. 8.12 - In Problems 15 to 18, use the given solutions of...Ch. 8.12 - In Problems 15 to 18, use the given solutions of...Ch. 8.12 - In Problems 15 to 18, use the given solutions of...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - In Problems 25 to 28, find a particular solution...Ch. 8.13 - In Problems 25 to 28, find a particular solution...Ch. 8.13 - In Problems 25 to 28, find a particular solution...Ch. 8.13 - In Problems 25 to 28, find a particular solution...Ch. 8.13 - If 10kg of rock salt is placed in water, it...Ch. 8.13 - A mass m falls under gravity (force mg ) through a...Ch. 8.13 - The acceleration of an electron in the electric...Ch. 8.13 - Suppose that the rate at which you work on a hot...Ch. 8.13 - Compare the temperatures of your cup of coffee at...Ch. 8.13 - A flexible chain of length l is hung over a peg...Ch. 8.13 - A raindrop falls through a cloud, increasing in...Ch. 8.13 - (a) A rocket of (variable) mass m is propelled by...Ch. 8.13 - The differential equation for the path of a planet...Ch. 8.13 - Use L15 and L31 to find the Laplace transform of...Ch. 8.13 - Use L32 and L9 to find the Laplace transform of t...Ch. 8.13 - Use the Laplace transform table to evaluate:...Ch. 8.13 - Use the Laplace transform table to evaluate:...Ch. 8.13 - Find the inverse Laplace transform of: p(p+a)3Ch. 8.13 - Find the inverse Laplace transform of: p2p2+a22Ch. 8.13 - Find the inverse Laplace transform of: 1p2+a23Ch. 8.13 - Prove the following shifting or translation...Ch. 8.13 - Use the table of Laplace transforms to find the...Ch. 8.13 - Solve Problems 47 and 48 either by Laplace...Ch. 8.13 - Solve Problems 47 and 48 either by Laplace...
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- Use the method of undetermined coefficients to solve the given nonhomogeneous system. dx dt = 2x + 3y − 8 dy dt = −x − 2y + 6 X(t) =arrow_forwardAs discussed in Section 8.3, the Markowitz model uses the variance of the portfolio as the measure of risk. However, variance includes deviations both below and above the mean return. Semivariance includes only deviations below the mean and is considered by many to be a better measure of risk. (a) Develop a model that minimizes semivariance for the Hauck Financial data given in the file HauckData with a required return of 10%. Assume that the five planning scenarios in the Hauck Financial Services model are equally likely to occur. Hint: Modify model (8.10)–(8.19). Define a variable ds for each scenario and let ds ≥ R − Rs with ds ≥ 0. Then make the objective function: Min 1 5 5 s = 1 ds2. Let FS = proportion of portfolio invested in the foreign stock mutual fund IB = proportion of portfolio invested in the intermediate-term bond fund LG = proportion of portfolio invested in the large-cap growth fund LV = proportion of portfolio invested in the large-cap value fund…arrow_forwardCalculus lll May I please have the blank lines completed, and final statement defined as a result? Thank you for the support!arrow_forward
- For each month of the year, Taylor collected the average high temperatures in Jackson, Mississippi. He used the data to create the histogram shown. Which set of data did he use to create the histogram? A 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92\ 55,\ 60,\ 64,\ 72,\ 73,\ 75,\ 77,\ 81,\ 83,\ 91,\ 91,\ 92 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92 B 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91\ 55,\ 57,\ 60,\ 65,\ 70,\ 71,\ 78,\ 79,\ 85,\ 86,\ 88,\ 91 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91 C 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93\ 55,\ 60,\ 63,\ 64,\ 65,\ 71,\ 83,\ 87,\ 88,\ 88,\ 89,\ 93 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93 D 55, 58, 60, 66, 68, 75, 77, 82, 86, 89, 91, 91\ 55,\ 58,\ 60,\ 66,\ 68,\ 75,\ 77,\ 82,\ 86,\ 89,\ 91,\ 91 55, 58, 60, 66, 68, 75, 77, 82, 86, 89, 91, 91arrow_forwardIn this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)arrow_forwardProblem 4. Margrabe formula and the Greeks (20 pts) In the homework, we determined the Margrabe formula for the price of an option allowing you to swap an x-stock for a y-stock at time T. For stocks with initial values xo, yo, common volatility σ and correlation p, the formula was given by Fo=yo (d+)-x0Þ(d_), where In (±² Ꭲ d+ õ√T and σ = σ√√√2(1 - p). дго (a) We want to determine a "Greek" for ỡ on the option: find a formula for θα (b) Is дго θα positive or negative? (c) We consider a situation in which the correlation p between the two stocks increases: what can you say about the price Fo? (d) Assume that yo< xo and p = 1. What is the price of the option?arrow_forward
- The Course Name Real Analysis please Solve questions by Real Analysisarrow_forwardWe consider a 4-dimensional stock price model given (under P) by dẴ₁ = µ· Xt dt + йt · ΣdŴt where (W) is an n-dimensional Brownian motion, π = (0.02, 0.01, -0.02, 0.05), 0.2 0 0 0 0.3 0.4 0 0 Σ= -0.1 -4a За 0 0.2 0.4 -0.1 0.2) and a E R. We assume that ☑0 = (1, 1, 1, 1) and that the interest rate on the market is r = 0.02. (a) Give a condition on a that would make stock #3 be the one with largest volatility. (b) Find the diversification coefficient for this portfolio as a function of a. (c) Determine the maximum diversification coefficient d that you could reach by varying the value of a? 2arrow_forwardQuestion 1. Your manager asks you to explain why the Black-Scholes model may be inappro- priate for pricing options in practice. Give one reason that would substantiate this claim? Question 2. We consider stock #1 and stock #2 in the model of Problem 2. Your manager asks you to pick only one of them to invest in based on the model provided. Which one do you choose and why ? Question 3. Let (St) to be an asset modeled by the Black-Scholes SDE. Let Ft be the price at time t of a European put with maturity T and strike price K. Then, the discounted option price process (ert Ft) t20 is a martingale. True or False? (Explain your answer.) Question 4. You are considering pricing an American put option using a Black-Scholes model for the underlying stock. An explicit formula for the price doesn't exist. In just a few words (no more than 2 sentences), explain how you would proceed to price it. Question 5. We model a short rate with a Ho-Lee model drt = ln(1+t) dt +2dWt. Then the interest rate…arrow_forward
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