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Assignment4-solution-public

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

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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 0 + 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 0 2 = xu 0 + u , y 00 2 = xu 00 + 2 u 0 , x 2 y 00 2 - xy 0 2 + y 2 = x 2 ( xu 00 + 2 u 0 ) - x ( xu 0 + u ) + xu = x 3 u 00 + x 2 u 0 = 0 derive x 2 on both sides, xu 00 + u 0 = 0 . Let w = u 0 , then xw 0 + w = 0 , This is a homogeneous linear equation, or separable equation, w = c/x where c is an arbitrary constant. Because we only need one solution, for
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