3.2 General Results for Linear Equations The linear equation of the nth order is Pn(t)y(t +...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
3.2 General Results for Linear Equations The linear equation of the nth order is Pn(t)y(t + n) + · + po(t)y(t) = r(t), (3.4) where po(t),..., Pn (t) and r(t) are assumed to be known and po(t) = 0, Pn(t) 0 for all t. If r(t) # 0, we say that Eq. (3.4) is "nonhomogeneous." As in Section 3.1, we will study Eq. (3.4) in association with the corresponding homogeneous equation Pn(t)u(t + n) + + po(t)u(t) = 0. (3.4') Note that Eq. (3.4) can also be written using the shift operator as (Pn(1) E" + + Po(t)Eº) y(t) = r(t), where EO 1. Since E = A + 1, it is also possible to write Eq. (3.4) in terms of the difference operator. However, the following example shows that the order of the equation is not apparent in that case. Theorem 3.8. Let ui(t),.. un (t) be independent solutions of Eq. (3.4'). Then y(t) = a₁(t)u₁(t) + ··· + an(t)un(t) is a solution of Eq. (3.4), provided a₁,, an satisfy the matrix equation Aai(t)" W(t + 1) Aan(t) 0 3.45 Use Theorem 3.8 to solve the first order equation y(t+1) - 2y(t) ²¹ (3). 2 3.2 General Results for Linear Equations The linear equation of the nth order is Pn(t)y(t + n) + · + po(t)y(t) = r(t), (3.4) where po(t),..., Pn (t) and r(t) are assumed to be known and po(t) = 0, Pn(t) 0 for all t. If r(t) # 0, we say that Eq. (3.4) is "nonhomogeneous." As in Section 3.1, we will study Eq. (3.4) in association with the corresponding homogeneous equation Pn(t)u(t + n) + + po(t)u(t) = 0. (3.4') Note that Eq. (3.4) can also be written using the shift operator as (Pn(1) E" + + Po(t)Eº) y(t) = r(t), where EO 1. Since E = A + 1, it is also possible to write Eq. (3.4) in terms of the difference operator. However, the following example shows that the order of the equation is not apparent in that case. Theorem 3.8. Let ui(t),.. un (t) be independent solutions of Eq. (3.4'). Then y(t) = a₁(t)u₁(t) + ··· + an(t)un(t) is a solution of Eq. (3.4), provided a₁,, an satisfy the matrix equation Aai(t)" W(t + 1) Aan(t) 0 3.45 Use Theorem 3.8 to solve the first order equation y(t+1) - 2y(t) ²¹ (3). 2
Expert Answer:
Answer rating: 100% (QA)
SOLUTION To solve the firstorder equation yt1 2yt 2t binomt5 using Theorem 38 we need to find the independent solutions of the corresponding homogeneo... View the full answer
Related Book For
Posted Date:
Students also viewed these accounting questions
-
For two samples from what are assumed to be normally distributed populations, the sample sizes and standard deviations are n1 = 10, s1 = 23.5, n2 = 9, and s2 = 10.4. At the 0.10 level of...
-
For two samples from what are assumed to be normally distributed populations, the sample sizes and standard deviations are n1 = 28, s1 = 103.1, n2 = 41, and s2 = 133.5. At the 0.10 level of...
-
When all cash flows are assumed to be discounted by a single yield, why is there no difference between the duration (sensitivity to yield), interest-rate duration (sensitivity to rates), and spread...
-
In Exercises find the second derivative of the function. f(x) = x sin x
-
The following accounts and balances were taken from the records of US Airways (dollars in millions). Identify each account as an income statement or balance sheet account. Where is each account...
-
a. Barga Company purchases $20,000 of equipment on January 1, 2015. The equipment is expected to last five years and be worth $2,000 at the end of that time. Prepare the entry to record one years...
-
What is the significance of the debt-to-equity ratio and how is it computed?
-
The Beal Manufacturing Company's costing system has two direct-cost categories: direct materials and direct manufacturing labor. Manufacturing overhead (both variable and fixed) is allocated to...
-
How does a Bureau of Alcohol, Tobacco, firearms and explosive support the general goals of criminal justice, what is an initiative plan to implement is it worth implement? Which key personnel,...
-
Arroy Snackfoods is considering replacing a five-year-old machine that originally cost $75,000. It was being depreciated using straight-line to an expected salvage value of zero over its original...
-
1 ) An object with an initial angular velocity of 6 rad / s undergoes a constant angular deceleration of 5 rad / s 2 . Calculate the angular displacement after 5 s . 2 ) A fan blade rotates with an...
-
PMF Random variables X and Y have the joint PX,Y (x, y) >-{8/ = 0, c|x+y], x= -2,0, 2; y = -1,0, 1, otherwise. What is the value of the constant c? What is P[Y X]? What is P[Y = X]? What is P[X < 1]?
-
Read the article You Have an Amazing New Healthcare Innovation, Now What? Legacy DNA. The article advises healthcare innovators to tout the innovative elements of their products or services loudly...
-
What distinguishes NoSQL databases from traditional SQL-based databases, and what are some common use cases for NoSQL ?
-
Evaluate each limit, if it exists. Show all steps. (2 marks each) b) a) lim 1 XII x +6x+8 +2x-8 1333 lim c) *-3x + 3 e) lim X+X+7 lim x+3 3 3 1 d) ***(x+1 lim x-16 f) lim x-16x-4
-
HiTech Corporation is considering investing in specialized machinery with a 2-year life that will generate significant cash savings during its life. The details related to this investment are:...
-
Discuss how COVID-19 transformed consumer spending habits. Provide an example. What are some procurement challenges procurement professionals face during COVID-19?
-
What is the amount of total interest dollars earned on a $5,000 deposit earning 6% for 20 years?
-
A casting machine is supposed to produce units for which the standard deviation of weights is 0.4 ounces. According to the maker of the machine, it should be shut down and thoroughly examined for...
-
Given the honeymoon expenditures for the 300 couples in Exercise 3.66 (file XR03066), generate a simple random sample consisting of 30 data values. In exercise Brides magazine reports the cost of the...
-
During fiscal 1995, Ashland Exploration drilled 7 wells in the United States and 3 in Nigeria. One of the wells in Nigeria was productive and 2 of the wells in the United States were productive. The...
-
Describe the changes that have taken place in share trading venues over the past decade.
-
You have been asked by the prospective directors of a shortly to be established business what is meant by ordinary shares, preference shares and debt capital. Further, you have been asked to provide...
-
Explain the acronyms AIM, SETS, MTF and HFT.
Study smarter with the SolutionInn App