I ATTACHED HOMEWORK 3 , and pictures from books shows the the equations. Also, I upload two scripts to edit them in order to solve the problem 5 and the extra credit question.
Homework 3: due 10/27 by 6pm The overall assignment is worth ten points. No partial credit is given, so check your work carefully. The first two problems are taken from the appendix of the textbook Foundations in Applied Nuclear Engineering Analysis by Glenn E. Sjoden (2nd Edition). From problem set #2 starting on p. 358. 1. (1 point) Question 4. From problem set #3 starting on p. 367. 2. (2 points total) Question 5. (Hint: use the variable substitution z = lnx). Note, the first edition of the textbook has a typo in this problem. The equation should read: x2 d2y dx2 + 4x dy dx + 2y = x cos (ln (x)) . One point for the homogeneous solution and one point for the particular solution. Additional problems: 3. (2 points total) By an appropriate change of variables, several equations can be written in the form of Bessel’s equation given by: x2 d2y dx2 + x dy dx + ( x2 − ν2 ) y = 0. (1) Using the variable substitutions indicated below, show the following equations can be rewritten in the form given by Eq. (1). Once the equation is in the standard form, find the general solution in terms of Bessel functions (no need to solve via Frobenius’ method, just identify the solution in terms of the appropriate Bessel functions) (a) (1 point) Use the substitution z = αx, x2 d2y dx2 + x dy dx + ( α2x2 − 1 4 ) y = 0, (b) (1 point) Use the substitution z = √ x, 4x2 d2y dx2 + 4x dy dx + ( x− 1 36 ) y = 0, 1 4. (2 points) Find the general solution to: d2y dx2 + 16y = 16 cos (4x) . One point for the homogenous solution, and one point for the particular solution. 5. (3 points total) Legendre polynomials satisfy the orthogonality relation:∫ 1 −1 dxPn (x)Pm (x) = 2 2n+ 1 δm,n. (2) (a) (1 point) Check Eq. (2) by analytically evaluating the integrals∫ 1 −1 dxP2 (x)P2 (x) and ∫ 1 −1 dxP2 (x)P1 (x). (b) (1 point) Write a code using the trapezoidal rule to numerically evaluate the integral: ∫ 1 −1 dxP6 (x)P0 (x) . (3) To an absolute tolerance of 10−6. How many intervals were required? Hint: You do not need to calculate Legendre polynomials by hand using Rodrigues’ formula. Their form can be looked up on several references (wikipedia, for example). (c) (1 point) Using Gaussian quadrature evaluate Eq. (3) exactly (i.e., up to roundoff error). What is the minimum number of quadrature points needed to obtain an exact result? Hint: When using Gaussian quadrature points and weights, make sure to use the exact values. (d) (1 point extra credit) The integral∫ 1 −1 dxP6 (x)P2 (x) . (4) vanishes identically with a 2-point quadrature rule. Why? Extra Credit (3 points): From problem set #3 starting on p. 367. Question 4 (1 point for the analytic solution and 2 points for the numerical solution). Use a 4th order Runge-Kutta method when solving the system numerically. 2 % Makes a nice plot % % clear all; % define the values on the horizontal axis numxpts = 100; xMin = -1.0; xMax = 1; xVec = xMin:(xMax-xMin)/(numxpts-1):xMax; % define the functions you want to plot f = xVec.^2; g = xVec.^2 - xVec.^3; % plot the functions hold on; plot(xVec,f,'b-','linewidth',2); hold off; hold on; plot(xVec,g,'r--','linewidth',2); hold off; % make the plot pretty xlabel('X','fontsize',22) ylabel('F(X)','fontsize',22) %xlim([xMin xMax]) %ylim([1 1.5]) set(gca,'FontSize',22) %set(gca,'xscale','log') %set(gca,'yscale','log')