Assignment5-solution-public

Assignment5-solution-public - Math 201 Assignment #5 Due:...

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Unformatted text preview: Math 201 Assignment #5 Due: November 1 in tutorial TAs: grade problems 2 and 3. 15 marks each. Give effort marks for 1, 4, 5, and 6, 5 marks each fi shown effort to finish, or 0 if left empty. 1. Solve the initial value problem 4 y 00- 4 y + y = 0 , y (0) = 1 , y (0) = 1 . Auxiliary equation: 4 2- 4 + 1 = (2 - 1) 2 = 0 , = 1 / 2 , repeated root y 1 = e x/ 2 , y 2 = xy 1 = xe x/ 2 General solution: y = c 1 y 1 + c 2 y 2 = c 1 e x/ 2 + c 2 xe x/ 2 Determine c 1 and c 2 using initial conditions: y (0) = c 1 = 1 , y ( x ) = c 1 2 e x/ 2 + c 2 e x/ 2 + c 2 x 2 e x/ 2 , y (0) = c 1 2 + c 2 = 1 , c 2 = 1 / 2 y = e x/ 2 + 1 2 xe x/ 2 2. Find the general solution of y 000 + y = xe x . Homogeneous part: y 000 + y = 0 Auxiliary equation: 3 + = ( 2 + 1) = 0 , = 0 , = i . Three independent solutions: * From = 0: y 1 = e 1 t = e t = 1 , * From = i , y 2 = e t cos t = cos t , y 3 = e t sin t = sin t . y c = c 1 y 1 + c 2 y 2 + c 3 y 3 = c 1 + c 2 cos t + c 3 sin t Particular solution: Guess for y p = ( ax + b ) e x y p = ae x + ( ax + b ) e x = ( ax + a + b ) e x , y 00 p = ae x + ( ax + a + b ) e x = ( ax + 2 a + b ) e x , y 000 p = ( ax + 3 a + b ) e x , y 000 p + y p = ( ax + 3 a + b )...
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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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Assignment5-solution-public - Math 201 Assignment #5 Due:...

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