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Tutorial6-solution

# Tutorial6-solution - Math 201 Fall 2011 Tutorial 6 Tuesday...

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Math 201 Fall 2011 Tutorial 6 Tuesday October 18 TAs: Points need to emphasize Order of polynomial If the guessed solution is a solution of the homogeneous equation, multiply it by x what to do if g ( x ) is a product of polynomials, sin and cos , and e ax what to do if g ( x ) is a product of polynomials, sin and cos , and e ax 1. Solve y 00 - 2 y 0 = 4 x + 2 . The general solution y g to y 00 - 2 y 0 = 0 , the auxiliary equation is λ 2 - 2 λ = 0 , λ 1 = 0 , λ 2 = 2 y 1 = e λ 1 t = e 0 x = 1 , y 2 = e λ 2 t = e 2 x So, y g = c 1 y 1 + c 2 y 2 = c 1 + c 2 e 2 x . A particular solution y p : because the right hand side is a polynomial, we guess for a polynomial solution y p with order n . Note that the left hand side has an order n - 1 , the right hans side has an order 1 , so n - 1 = 1 , n = 2 , that is, we guess for a quadratic solution, y p = ax 2 + bx + c , y 00 p - 2 y 00 p = 2 a - 2(2 ax + b ) = - 4 ax + 2 a - 2 b = 4 x + 2 compare the coefficients of the polynomials on both sides, - 4 a = 4 , 2 a - 2 b = 2 , so, a = - 1 , b = - 1 , c is undetermined, a free parameter, pick c = 0 for simplification.

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Tutorial6-solution - Math 201 Fall 2011 Tutorial 6 Tuesday...

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