d2x K dt m - 1 (d+ x) - 1 (d- x) (a) Classify this ODE...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
d²2x K dt² m - 1 (d+ x)² - 1 (d- x)² (a) Classify this ODE and explain why there is little hope of solving it as is. (b) In order to solve, let's assume x < d and then replace the right-hand side function with an appropriate Taylor series approximation (you will do this in parts (c) and (d)). Explain what the condition x < d means physically. (c) We want to expand the right-hand side function in an appropriate Taylor series. What is the "appropriate" Taylor series? Let the variable that we are expanding in be called z. What quantity is playing the role of z? And are we expanding around z = 0 (Maclaurin series) or some other value of z? [HINT: factor a ď² out of the denominator of both terms.] Also, how many terms in the series do we need to keep? [HINT: we are trying to simplify the ODE. How many terms in the series do you need in order to make the ODE look like an equation that you know how to solve?] (d) Expand the right-hand side function of the ODE in the appropriate Taylor series you described in part (c). [You have two options here. One is the "direct" approach. The other is to use one series to obtain a different series via re-expanding, as you did in class for x²/3. Pick one and do it. If you feel up to the challenge, do it both ways and make sure they agree.] (e) If all went well, your new, approximate ODE should resemble the simple harmonic oscillator equation. What is the frequency of oscillations of the solutions to that equation in terms of k, m, and d? d²2x K dt² m - 1 (d+ x)² - 1 (d- x)² (a) Classify this ODE and explain why there is little hope of solving it as is. (b) In order to solve, let's assume x < d and then replace the right-hand side function with an appropriate Taylor series approximation (you will do this in parts (c) and (d)). Explain what the condition x < d means physically. (c) We want to expand the right-hand side function in an appropriate Taylor series. What is the "appropriate" Taylor series? Let the variable that we are expanding in be called z. What quantity is playing the role of z? And are we expanding around z = 0 (Maclaurin series) or some other value of z? [HINT: factor a ď² out of the denominator of both terms.] Also, how many terms in the series do we need to keep? [HINT: we are trying to simplify the ODE. How many terms in the series do you need in order to make the ODE look like an equation that you know how to solve?] (d) Expand the right-hand side function of the ODE in the appropriate Taylor series you described in part (c). [You have two options here. One is the "direct" approach. The other is to use one series to obtain a different series via re-expanding, as you did in class for x²/3. Pick one and do it. If you feel up to the challenge, do it both ways and make sure they agree.] (e) If all went well, your new, approximate ODE should resemble the simple harmonic oscillator equation. What is the frequency of oscillations of the solutions to that equation in terms of k, m, and d?
Expert Answer:
Answer rating: 100% (QA)
The given ordinary differential equation ODE is dxdt 777dx2 1dx2 This ODE is a firstorder nonlinear ... View the full answer
Related Book For
Posted Date:
Students also viewed these accounting questions
-
How does the water-diamond paradox explain why there is such a poor correlation between the price of a good and the total utility a person receives from it?
-
Explain why there is an inverse relationship between planned detection risk and the amount of evidence an auditor collects for a specific audit objective.
-
Explain why there is a different money multiplier for each definition of the money supply.
-
Evaluate the limit or determine that it does not exist. |xx| lim (x,y)(0,0) [x] + [yl
-
Factory Direct manufactures and sells manufactured homes. Traditionally, it has distributed indirect costs to its three construction plants based on materials cost. Each plant builds different models...
-
Which of the following is a valid header for a procedure that receives an integer followed by a number with a decimal place? a. Private Sub GetFee(intBase As Value, decRate As Value) b. Private Sub...
-
There is a market for bets on the outcome of a coin toss. The possible outcomes are heads, tails, and edge. There are three assets traded in that market: Asset A pays \(\$ 1\) independent of the...
-
The Hernandez Manufacturing Company has two producing departments, machining and assembly. Mr. Hernandez recently automated the machining department. The installation of a CAM system, together with...
-
Define moral hazard. Does it apply only to consumers? If your demand for health care services is more or less elastic, will there be more or less (potential) moral hazard effect for a given...
-
Howies Bakery is one of the most popular bakeries in town, and the favorite at Howies is French bread. Each day of the week, Howies bakes a number of loaves of French bread, more or less according to...
-
Based on the given information ,where X ={1,2,3,4,5 } and Y={a,b,c} and the mapping F:X->Y is as below: Element 1 and 3 in X mapped to a. Element 2 and 5 in X mapped to b. Element 4 in X is mapped...
-
A hospitality compOne of the principal and most welcome advantages of a computerized accounting system is its capability to automatically:any is the maker of an $18,000 note to be paid in quarterly...
-
Compare and contrast the gram negative, gram positive, acid fast and, archaea cell walls. with Example
-
Differentiate between plasma membrane of archaea, bacteria and eukaryote structure?
-
Excluding securitization, as it relates to the disintermediation of bank loans, what is securitization? Why would a firm (bank or company) want to securitize an asset? Explain briefly
-
How would you value a company that has been in business for 6-months? You are provided income and balance sheet statements for the firm. Lists the steps that you would follow. How would your...
-
Assume that a project earns $17min in each of the next two years (cash flows are received at the end of each year). Then in year three the cash inflows are 3% higher compared to the previous year,...
-
Do animals have rights? If so, what are they? What duties do human beings have toward animals? Does KFC protect animal welfare at an acceptable level?
-
The magnetic field at the center of a 50-turn coil of radius 15 cm is 0.80 mT. Find the current in the coil.
-
The acceleration as a function of time of a massspring system is given by a = (0.60 m/s2) sin[(2 rad/s)t]. If the spring constant is 10 N/m, what are (a) The amplitude, (b) The initial velocity and...
-
Roughly speaking, the observable mass in our universe is all attributed to stars and gas clouds in the galaxies. (a) Assuming that each galaxy contains the mass of 200 -billion Suns and there are 200...
-
The following data from Lyre Ltd's accounts relates to two assets at 30 June 2023. At 30 June 2023 Lyre Ltd decides to adopt the revaluation model for both these assets. On this date land has a fair...
-
On 1 January 2023, Blue Jay Ltd revalued land from \(\$ 300000\) to \(\$ 500000\). On 1 January 2024, the company subsequently revalued the land to \(\$ 420000\). And on 1 January 2025 , the company...
-
Seagull Ltd purchased land for use as its corporate headquarters. A small factory that was on the land when it was purchased was torn down before construction of the office building began....
Study smarter with the SolutionInn App