Reduction of Order and Variation of parameters Consider the variable coefficient linear second order non-homogeneous ODE...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Reduction of Order and Variation of parameters Consider the variable coefficient linear second order non-homogeneous ODE ry" + 2(x - 1)y + (x-2)y=e ², x>0. 1. Write down the associated homogeneous equation. 2. A solution of the associated homogeneous equation is y₁ = e. Use the formula in the course notes for the method of reduction of order to find a second solution, 32, of the associated homogeneous equation. You MUST express the second solution in its simplest form (1) 32 = x³, where m and 3 are constants that you need to determine. Hint: Re-write the associated ODE in standard form to help with identifying the terms in the formula. 2 3. Use the Wronskian test to determine if y₁ and y2 are linearly independent for a > 0. 4. Write down the solution yn (r) of the associated homogeneous equation. 6 5. Use the formulas in the course notes for the method of variation of parameters to find a particular solution yp(x) of equation (1). Hint: Re-write the ODE in standard form in order to identify the non-homogeneous term in the formulas. 6. Write down the general solution to equation (1). 7. Find the solution to equation (1) that satisfies the initial conditions y(1) = 1 and (1) = -1. 8. Use Matlab to plot the solution to the IVP in 7 for 0 < < 5. Attach the plot to your assignment and describe the solution behaviour as a →∞. Reduction of Order and Variation of parameters Consider the variable coefficient linear second order non-homogeneous ODE ry" + 2(x - 1)y + (x-2)y=e ², x>0. 1. Write down the associated homogeneous equation. 2. A solution of the associated homogeneous equation is y₁ = e. Use the formula in the course notes for the method of reduction of order to find a second solution, 32, of the associated homogeneous equation. You MUST express the second solution in its simplest form (1) 32 = x³, where m and 3 are constants that you need to determine. Hint: Re-write the associated ODE in standard form to help with identifying the terms in the formula. 2 3. Use the Wronskian test to determine if y₁ and y2 are linearly independent for a > 0. 4. Write down the solution yn (r) of the associated homogeneous equation. 6 5. Use the formulas in the course notes for the method of variation of parameters to find a particular solution yp(x) of equation (1). Hint: Re-write the ODE in standard form in order to identify the non-homogeneous term in the formulas. 6. Write down the general solution to equation (1). 7. Find the solution to equation (1) that satisfies the initial conditions y(1) = 1 and (1) = -1. 8. Use Matlab to plot the solution to the IVP in 7 for 0 < < 5. Attach the plot to your assignment and describe the solution behaviour as a →∞.
Expert Answer:
Answer rating: 100% (QA)
To solve the given secondorder nonhomogeneous ODE well go through the steps of reduction of order and variation of parameters Lets tackle each part st... View the full answer
Related Book For
Essentials of Business Analytics
ISBN: 978-1285187273
1st edition
Authors: Jeffrey Camm, James Cochran, Michael Fry, Jeffrey Ohlmann, David Anderson, Dennis Sweeney, Thomas Williams
Posted Date:
Students also viewed these mathematics questions
-
Case Study: Quick Fix Dental Practice Technology requirements Application must be built using Visual Studio 2019 or Visual Studio 2017, professional or enterprise. The community edition is not...
-
The following additional information is available for the Dr. Ivan and Irene Incisor family from Chapters 1-5. Ivan's grandfather died and left a portfolio of municipal bonds. In 2012, they pay Ivan...
-
List three specific parts of the Case Guide, Objectives and Strategy Section (See below) that you had the most difficulty understanding. Describe your current understanding of these parts. Provide...
-
Explain how the structure of the cash flow statement represents another subsection of the accounting equation.
-
You are the manager of a monopoly that sells a product to two groups of consumers in different parts of the country. Group 1s elasticity of demand is 2, while group 2s is 6. Your marginal cost of...
-
(a) What is the difference between the pressure just below the water surface in a pool and the pressure in the air just above it? (b) Is the pressure inside a raindrop greater than, equal to, or...
-
List the items which the Companies Act 1985 requires auditors to include in their reports to shareholders.
-
In Table 7.4 on page 173, all ten top hitters in the major league baseball in 2011 had lower batting averages in 2012, supporting regression toward the mean. Treating averages as whole numbers...
-
According to the theory of strength, the bending moment of the hyperstatic beam at the central support is Mt = 27.5625 MNm . Solve the support reactions of the beam and draw the bending moment...
-
A supplier shipped $3,000 of ingredients on 12/29/17. Peyton receives an invoice for the goods, as well as a bill for freight for $175, all dated 12/29/17. Goods were shipped FOB suppliers warehouse....
-
Coop Inc. owns 40% of Chicken Inc., both Coop and Chicken are corporations. Chicken pays Coop a dividend of $10,000 in 2014. Chicken also reports financial accounting earnings of $20,000 for that...
-
Explain the uncertainty analysis of DSA, PSA, and Reliability parameters and their advantages.
-
What do you understand by the term insider trading? Explain the legal provisions in this regard with examples.
-
Explain the different ways in which finance companies act as financial intermediaries. Name the different types of finance companies.
-
What is financial engineering? How does it help a business firm?
-
What are the main characteristics of exchangetraded mutual funds? How are they different from no-load funds?
-
An investor has initial wealth w = 2, and a utility function U(w) - 1- exp(-Aw). A lottery has a payoff of 1 with probability p and a payoff of -1 with probability (1-p). (a) Suppose A = 1 and p =...
-
In Problem 8.43, determine the smallest value of for which the rod will not fall out of the pipe. IA -3 in.-
-
Consider again the CHB Inc. problem described in Problem 10. Suppose only a limted number of PPBs can be placed. CHB would like to place this limited number of PPBs in counties so that the allowable...
-
In preparing for the upcoming holiday season, Fresh Toy Company (FTC) designed a new doll called The Dougie that teaches children how to dance. The fixed cost to produce the doll is $100,000. The...
-
The Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form a. In this example, assume c0 = 5, c1 = 0.25, and c2 =...
-
Use a stem-and-leaf plot to display the data, which represent the runs scored by a batsman in a World Cup series. Organize the data using the indicated type of graph. Describe any patterns. 70 75 71...
-
Use a stem-and-leaf plot to display the data shown in the table at the left, which represent the drunk driving cases registered at 30 strategic road intersections. Organize the data using the...
-
Use a stem-and-leaf plot that has two rows for each stem to display the data, which represent the incomes (in millions) of the top 30 highest-paid tech CEOs. Organize the data using the indicated...
Study smarter with the SolutionInn App