A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Chapter III, Problem 2E
a.
To determine
To find the value of
b.
To determine
To find the value of
c.
To determine
To find the value of
d.
To determine
To find the value of
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1. [10 points] Given y₁(x) = x²² is a solution to the differential equation x²y"+6xy'+6y=0 (x>0), find a
second linearly independent solution using reduction of order.
>tt 1:32
> trend.1m 1m (sales
> summary(trend.1m)
-
tt) #3###23 (i) ####
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2107.220
57.997 36.332e-16 ***
tt
-43.500
3.067 -14.18 7.72e-15 ***
> trend = ts (fitted (trend.1m), start-start (sales), freq-frequency (sales))
sales trend ###23%23 (ii) ####
as.numeric((1:32 %% 4)
> X
> q1
> q2
> q3
> 94
=
=
=
=
-
as.numeric((1:32 %% 4)
as.numeric((1:32 %% 4)
as.numeric((1:32 %% 4)
== 1)
2)
==
== 3)
==
0)
> season.lm = 1m (resid (trend.1m) 0+q1 + q2 + q3 + q4) #3##23%23 (iii) ####
> summary(season.1m)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
q1
-38.41
43.27 -0.888 0.38232
92
18.80
43.27
0.435 0.66719
q3
-134.78
43.27
-3.115 0.00422 **
94
154.38
43.27 3.568
0.00132 **
> season = ts (fitted (season.lm), start=start (sales), freq=frequency (sales))
> Y X season %23%23%23%23 (iv) ####
>ar (Y, aic=FALSE, order.max=1) #23%23%23%23 (v) ####
Coefficients:
1
0.5704
Order selected 1 sigma 2 estimated as 9431
> ar(Y, aic=FALSE,…
Chapter III Solutions
A Transition to Advanced Mathematics
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- Refer to page 52 for solving the heat equation using separation of variables. Instructions: • • • Write the heat equation in its standard form and apply boundary and initial conditions. Use the method of separation of variables to derive the solution. Clearly show the derivation of eigenfunctions and coefficients. Provide a detailed solution, step- by-step. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 20 for orthogonalizing a set of vectors using the Gram-Schmidt process. Instructions: • Apply the Gram-Schmidt procedure to the given set of vectors, showing all projections and subtractions step-by-step. • Normalize the resulting orthogonal vectors if required. • Verify orthogonality by computing dot products between the vectors. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 54 for solving the wave equation. Instructions: • Apply d'Alembert's solution method or separation of variables as appropriate. • Clearly show the derivation of the general solution. • Incorporate initial and boundary conditions to obtain a specific solution. Justify all transformations and integrations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440AZF/view?usp=sharing]arrow_forward
- Refer to page 14 for calculating eigenvalues and eigenvectors of a matrix. Instructions: • Compute the characteristic polynomial by finding the determinant of A - XI. • Solve for eigenvalues and substitute them into (A - I) x = 0 to find the eigenvectors. • Normalize the eigenvectors if required and verify your results. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardExilet x = {a,b.c}dex.x―R> d(a,b) = d(b, c)=1' d(a, c) = 2 d(xx)=0VXEX is (x.d) m.s or not? 3.4 let x= d ((x,y), (3arrow_forwardHiw Show that sup (0,1) = 1 الفصل الثاني * Dif: let {an} be Seq. then fan?arrow_forward
- Please show as much work as possible to clearly show the steps you used to find each solution. If you plan to use a calculator, please be sure to clearly indicate your strategy. 1. The probability of a soccer game in a particular league going into overtime is 0.125. Find the following: a. The odds in favour of a game going into overtime. b. The odds in favour of a game not going into overtime. c. If the teams in the league play 100 games in a season, about how many games would you expect to go into overtime?arrow_forwardThe probability of being born in a particular month is about 1:12. Determine the probability of not being born in September. Express this ratio as a fraction, a decimal, a percent and in words.arrow_forwardIn his first hockey game of the season, Brayden takes a total of 10 shots on the goalie and scores 1 time. Later in the season, he takes 30 shots in total on the goalie. How many goals would you expect him to make? What assumptions are making? Are your assumptions realistic? Explain.arrow_forward
- The probability of being born in a particular month is about 1:12. Determine the probability of not being born in September. Express this ratio as a fraction, a decimal, a percent and in words.arrow_forwardThe probability of being born in a particular month is about 1:12. Determine the probability of not being born in September. Express this ratio as a fraction, a decimal, a percent and in words.arrow_forwardDevon is expected to receive 70% of the votes at the student council election. If there are 650 students in his school, how many are expected to vote for him?arrow_forward
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