Assignment4-solution-public

Assignment4-solution-public - Math 201 Assignment #4...

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Unformatted text preview: Math 201 Assignment #4 October 18, 2011 Total 30 marks: • Grade problems 1 and 4, 10 marks each, • 10 effort marks, 5 marks each for problems 2 and 3 (0 for finished, 5 for empty, but their correctness will bot be checked). 1. Consider the equation x 2 y 00- xy + y = 0 . (a) Find a solution y 1 in the form of y = x r (where r is a constant). • Substitute y = x r into the equation, x 2 r ( r- 1) x r- 2- xrx r- 1 + x r = ( r 2- 2 r + 1) x r = 0 • Note that because x 2 is the coefficient of y 00 , this equation requires that x 2 6 = 0 , i.e., x 6 = 0 . • Thus, r 2- 2 r + 1 = ( r- 1) 2 = 0 , r = 1 . So, y = x is a solution. (b) Use reduction of order to find a solution y 2 that is linearly independent to y 1 . (Evaluate the Wronskian and show that y 1 and y 2 are linearly independent). • Let y 2 = y 1 u = xu , substitute into the equation, note that y 2 = xu + u , y 00 2 = xu 00 + 2 u , x 2 y 00 2- xy 2 + y 2 = x 2 ( xu 00 + 2 u )- x ( xu + u ) + xu = x 3 u 00 + x 2 u = 0 derive...
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This note was uploaded on 01/15/2012 for the course MATH 201 taught by Professor Steacy during the Winter '10 term at University of Victoria.

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Assignment4-solution-public - Math 201 Assignment #4...

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