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# test3-0 - MATH 315 FALL 1998 TEST 3 SOLUTION KEY 1(10 pts...

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MATH 315 FALL 1998 TEST 3 SOLUTION KEY 1. (10 pts.) Use Power series to solve the differential equation y ' + y = 3 x , with y (0) = 1. (a) Find the recursion formula for the coefficients a n in the power series representation of the solution . Answer: Substituting y ( x ) into the equation , so , or . Therefore a 1 + a 0 = 0, 2 a 2 + a 1 =3, and ( n +1) a n +1 + a n = 0 for n >1. The recursion formula is a n +1 = - a n /( n +1), for n >1. (b) Use this recursion formula to determine a general formula for the coefficient a n . Answer: Using y (0)=1, a 0 = 1, so a 1 = -1, a 2 = (3+1)/2=2, a 3 =-2/3, , and therefore a n = (-1) n 4/ n !, for n >1. 2. (10 pts.) Determine the Laplace transform for the function Answer: 3. (20 pts.) Use Power series to solve the differential equation y '' + 4 y = 0 , with y (0) = 0, y '(0)=1. (a) Find the recursion formula for the coefficients a n in the power series representation of the solution .

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Answer: Substituting y ( x ) into the equation , so , or . Therefore ( n +1)( n +2) a n +2 +4 a n =0 for , and the recursion formula is a n +2 = -4 a n /(( n +1)( n +2)), for . (b) Use this recursion formula to determine a general formula for the coefficient
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test3-0 - MATH 315 FALL 1998 TEST 3 SOLUTION KEY 1(10 pts...

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