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exam paper - AMA 262 THE UNIVERSITY OF SHEFFIELD SCHOOL OF...

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1 AMA262 THE UNIVERSITY OF SHEFFIELD SCHOOL OF MATHEMATICS AND STATISTICS May/June 2006 2 hours APPLIED DIFFERENTIAL EQUATIONS Marks will be awarded for your best FOUR answers. 1.(i) A procedure for obtaining the numerical solution of the differential equation is given as where and where are constants. Show that the conditions are sufficient to ensure that the procedure is second-order. (12 marks) Question 1 continued on page 2 AMA262 TURN OVER
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2 AMA262 Question 1 continued from page 1 (ii) Use the results of part (i) to confirm that the procedure where is second-order and use it with to estimate given that and . Work correct to 4 decimal places. (7 marks) (iii) Given that the interval of absolute stability for the Euler 1 method applied to the test equation is , determine an upper limit on the value of for the Euler 1 method to be absolutely stable when applied to the equation over the interval . (6 marks) AMA262 CONTINUED
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3 AMA262 2. Derive the first three non-zero terms of the Taylor series solution
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