Q1. (6 marks)
a) Find all eigenvalues for the matrix XXXXXXXXXXmarks)
) Find all eigenvectors for the same matrix. Check your solution. (2 marks)
c) Compute (Hint: Homework sheet 3, Question XXXXXXXXXXmarks)
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Q2. (9 marks)
a) Determine the Maclaurin series for XXXXXXXXXXmarks)
) Find the associated radius of convergence for the Maclaurin series in a) (3 marks)
Write your series using notation. (Hints: lecture 11 Interval and Radius of convergence; lecture 14 Review: Maclaurin series)
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Q3. (6 marks)
Test the following series for convergence or divergence
(Hint: lecture 14 Integral Test; Worksheet 13 Q2c)
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Q4. (6 marks)
Find the general solution of the following differential equation (and check your answers by differentiating)
Note that is a function of ; is the first derivative, and is the second derivative.
(Hints: lecture 16 Separable DE; Worksheet 16 Q3.)
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Q5 (8 marks)
Find the solution to the following initial-value problems
,
(Hints: Worksheet 18 Q3, Q4(b) and Q4 (e)
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