Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Chapter 6.2, Problem 1E
Using initial condition
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πT
| cos x |³ dx
59
2
2. Consider the ODE
u' = ƒ (u) = u² + r
where r is a parameter that can take the values r = −1, −0.5, -0.1, 0.1. For each value of r:
(a) Sketch ƒ(u) = u² + r and determine the equilibrium points.
(b) Draw the phase line.
(d) Determine the stability of the equilibrium points.
(d) Plot the direction field and some sample solutions,i.e., u(t)
(e) Describe how location of the equilibrium points and their stability change as you increase the
parameter r.
(f) Using the matlab program phaseline.m generate a solution for each value of r and the initial
condition u(0) = 0.9. Print and turn in your result for r = −1. Do not forget to add a figure caption.
(g) In the matlab program phaseline.m set the initial condition to u(0) = 1.1 and simulate the ode
over the time interval t = [0, 10] for different values of r. What happens? Why? You do not need to
turn in a plot for (g), just describe what happens.
The following are suggested designs for group sequential studies. Using PROCSEQDESIGN, provide the following for the design O’Brien Fleming and Pocock.• The critical boundary values for each analysis of the data• The expected sample sizes at each interim analysisAssume the standardized Z score method for calculating boundaries.Investigators are evaluating the success rate of a novel drug for treating a certain type ofbacterial wound infection. Since no existing treatment exists, they have planned a one-armstudy. They wish to test whether the success rate of the drug is better than 50%, whichthey have defined as the null success rate. Preliminary testing has estimated the successrate of the drug at 55%. The investigators are eager to get the drug into production andwould like to plan for 9 interim analyses (10 analyzes in total) of the data. Assume thesignificance level is 5% and power is 90%.Besides, draw a combined boundary plot (OBF, POC, and HP)
Chapter 6 Solutions
Numerical Analysis
Ch. 6.1 - Show that the function y(t)=tsint is a solution of...Ch. 6.1 - Show that the function y(t)=esint is a solution of...Ch. 6.1 - Use separation of variables to find solutions of...Ch. 6.1 - Find the solutions of the IVP given by y(0)=0 and...Ch. 6.1 - Apply Eulers Method with step size h=1/4 to the...Ch. 6.1 - Apply Eulers Method with step size h=1/4 to the...Ch. 6.1 - (a) Show that y=tan(t+c) is a solution of the...Ch. 6.1 - (a) Show that y=tanh(t+c) is a solution of the...Ch. 6.1 - For which of these initial value problems on [0,...Ch. 6.1 - Sketch the slope field of the differential...
Ch. 6.1 - Find the solutions of the initial value problems...Ch. 6.1 - (a)Show that if a0, the solution of the initial...Ch. 6.1 - Use separation of variables to solve the initial...Ch. 6.1 - Find the solution of the initial value problem...Ch. 6.1 - Prob. 15ECh. 6.1 - Prob. 16ECh. 6.1 - Prob. 17ECh. 6.1 - Apply Eulers Method with step size h=0.1 on [0, 1]...Ch. 6.1 - Plot the Eulers Method approximate solutions for...Ch. 6.1 - Plot the Eulers Method approximate solutions for...Ch. 6.1 - Prob. 4CPCh. 6.1 - For the IVPs in Exercise 4, make a log-log plot of...Ch. 6.1 - Prob. 6CPCh. 6.1 - Plot the Eulers Method approximate solution on [0,...Ch. 6.1 - Plot the Eulers Method approximate solution on [0,...Ch. 6.1 - Calculate the Eulers Method approximate solution...Ch. 6.1 - Calculate the Eulers Method approximate solution...Ch. 6.1 - Plot the Eulers Method approximate solution on [0,...Ch. 6.2 - Using initial condition y(0)=1 and step size...Ch. 6.2 - Using initial condition y(0)=0 and step size...Ch. 6.2 - Find the formula for the second-order Taylor...Ch. 6.2 - Apply the second-order Taylor Method to the...Ch. 6.2 - (a) Prove (6.22) (b) Prove (6.23).Ch. 6.2 - Apply the Explicit Trapezoid Method on a grid of...Ch. 6.2 - Prob. 2CPCh. 6.2 - Prob. 3CPCh. 6.2 - Prob. 4CPCh. 6.2 - Prob. 5CPCh. 6.2 - Plot the Trapezoid Method approximate solution on...Ch. 6.2 - Calculate the Trapezoid Method approximate...Ch. 6.2 - Calculate the Trapezoid Method approximate...Ch. 6.2 - Prob. 9CPCh. 6.3 - Apply the Eulers Method with step size h=1/4 to...Ch. 6.3 - Apply the Trapezoid Method with h=1/4 to the...Ch. 6.3 - Convert the higher-order ordinary differential...Ch. 6.3 - Apply the Trapezoid Method with h=1/4 to the...Ch. 6.3 - (a) Show that y(t)=(et+ett2)/21 is the solution of...Ch. 6.3 - Apply Eulers Method with step sizes h=0.1 and 0.01...Ch. 6.3 - Carry out Computer Problem 1for the Trapezoid...Ch. 6.3 - Prob. 3CPCh. 6.3 - Prob. 4CPCh. 6.3 - Prob. 5CPCh. 6.3 - Adapt pend.m to build a damped pendulum with...Ch. 6.3 - Prob. 7CPCh. 6.3 - Prob. 8CPCh. 6.3 - Prob. 9CPCh. 6.3 - Prob. 10CPCh. 6.3 - Prob. 11CPCh. 6.3 - Prob. 12CPCh. 6.3 - Prob. 13CPCh. 6.3 - Prob. 14CPCh. 6.3 - Prob. 15CPCh. 6.3 - A remarkable three-body figure-eight orbit was...Ch. 6.4 - Apply the Midpoint Method for the IVPs...Ch. 6.4 - Carry out the steps of Exercise 1 for the IVPs...Ch. 6.4 - Apply fourth-order Runge-Kutta Method to the IVPs...Ch. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.4 - Consider the initial value problem y=y . The...Ch. 6.4 - Prob. 7ECh. 6.4 - Prob. 1CPCh. 6.4 - Apply the fourth-order Runge-Kutta Method solution...Ch. 6.4 - Carry out the steps of Computer Problem 2, but...Ch. 6.4 - Prob. 4CPCh. 6.4 - Plot the fourth-order Runge-Kutta Method...Ch. 6.4 - Plot the fourth-order Runge-Kutta Method...Ch. 6.4 - Prob. 7CPCh. 6.4 - Prob. 8CPCh. 6.4 - Prob. 9CPCh. 6.4 - Prob. 10CPCh. 6.4 - Adapt the orbit .m MATLABs program to animate a...Ch. 6.4 - Assess the conditioning of the Lorenz equations by...Ch. 6.4 - Follow two trajectories of the Lorenz equations...Ch. 6.4 - Prob. 14CPCh. 6.4 - Prob. 15CPCh. 6.4 - Prob. 16CPCh. 6.4 - Prob. 17CPCh. 6.4 - Prob. 18CPCh. 6.4 - Run tacoma.m with wind speed W=80km/hr and initial...Ch. 6.4 - Replace the Trapezoid Method by fourth-order...Ch. 6.4 - The system is torsionally stable for W=50km/hr ....Ch. 6.4 - Find the minimum wind speed W for which a small...Ch. 6.4 - Prob. 5SACh. 6.4 - Prob. 6SACh. 6.4 - Prob. 7SACh. 6.5 - Write a MATLAB implementation of RK23 (Example...Ch. 6.5 - Prob. 2CPCh. 6.5 - Prob. 3CPCh. 6.5 - Compare the results of Computer Problem 3 with the...Ch. 6.5 - Apply a MATLAB implementation of RKF45 to...Ch. 6.6 - Using initial condition y(0)=0 and step size...Ch. 6.6 - Find all equilibrium solutions and the value of...Ch. 6.6 - Prob. 3ECh. 6.6 - Consider the linear differential equation y=ay+b...Ch. 6.6 - Apply Backward Euler, using Newtons Method as a...Ch. 6.6 - Carry out the steps in Computer Problem1 for the...Ch. 6.7 - Apply the Adams-Bashforth Two-Step Method to the...Ch. 6.7 - Carry out the steps of Exercise 1 on the IVPs...Ch. 6.7 - Prob. 3ECh. 6.7 - Prob. 4ECh. 6.7 - Show that the Implicit Trapezoid Method (6.89) is...Ch. 6.7 - Prob. 6ECh. 6.7 - Prob. 7ECh. 6.7 - Prob. 8ECh. 6.7 - Find the order and stability type for the...Ch. 6.7 - Prob. 10ECh. 6.7 - Prob. 11ECh. 6.7 - The Mime-Simpson Method is a weakly stable...Ch. 6.7 - Prob. 13ECh. 6.7 - (a) Use the matrix formulation to find the...Ch. 6.7 - Prob. 15ECh. 6.7 - (a) Use the matrix formulation to find the...Ch. 6.7 - Adapt the exmultistep.m program to apply the...Ch. 6.7 - Adapt the exmultistep.m program to apply the...Ch. 6.7 - Prob. 3CPCh. 6.7 - Prob. 4CPCh. 6.7 - Prob. 5CPCh. 6.7 - Prob. 6CPCh. 6.7 - Prob. 7CPCh. 6.7 - Prob. 8CPCh. 6.7 - Prob. 9CPCh. 6.7 - Change Program 6.8 into a fourth-order...
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