(F)prelim1_test&solUnknown

(F)prelim1_test&solUnknown - PRELIM 1 MATH 2930 CORNELL...

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PRELIM 1 MATH 2930 CORNELL UNIVERSITY SOLUTIONS (20%) 1. Let a be some unknown constant. Consider y 00 + 2 ay 0 + ( a 2 - 1) y = 0 i) Solve the general differential equation. ii) Solve the I.V.P. if y (0) = 1, y 0 (0) = 0. Solution. i) The characteristic equation is x 2 + 2 ax + ( a 2 - 1) = 0, so that ( x + a - 1)( x + a + 1) = 0 so that r = - a ± 1 are the roots. Then the solution is y = C 1 e ( - a +1) t + C 2 e ( - a - 1) t . ii) Plugging in 0 for t and setting equal to y (0) = 1 (given) we get C 1 + C 2 = 1. Similarly, plugging in 0 into y 0 and setting equal to 0 gives ( - a + 1) C 1 + ( - a - 1) C 2 = 0. Solving gives C 1 = 1+ a 2 , C 2 = 1 - a 2 , so our particular solution becomes y 0 ( t ) = 1 + a 2 e ( - a +1) t + 1 - a 2 e ( - a - 1) t (20%) 2. Solve the differential equation: y 0 + y 2 +1 y e - x = 0. Solution. This is a separable equation. To separate we want to mul- tiply through by y y 2 +1 , and move the e - x (and the dx ) to the right hand side to get y p y 2 + 1 dy = - e - x dx. Integrating gives p y 2 + 1 = e - x + C, so that y = ± p ( e - x + C ) 2 - 1 gives the solutions.
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