# exam2 - Midterm 2 of Math 255.01(Wi11 1:30pm-2:18pm The...

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Midterm 2 of Math 255.01 (Wi11) February 25, 2011 The Ohio State University 1:30pm-2:18pm Including the cover sheer, this exam consists of ﬁve (5) pages and four (4) problems and is worth a total of 100 points. The point value of each problem is indicated. To obtain full credit, you must have the correct answers along with relevant supporting work to jus- tify them. Partial credit will be given based on the work that is shown. However, answers without sufﬁcient supporting work will receive no credit. Also, you must complete the exam in 48 minutes. Good luck! Name: Signature: OSU Internet Username: Problem # Score 1 (30 pts) 2 (30 pts) 3 (15 pts) 4 (25 pts)

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1. Suppose that y 1 ( x ) = 1 + x 2 and y 2 ( x ) are solutions to the equation (1 - x 2 ) d 2 y dx 2 + 2 x dy dx - 2 y = 0 , - 1 < x < 1 , such that y 2 (0) = 0 and y 0 2 (0) = 2 . (a) (12) Use Abel’s theorem to calculate the Wronskian W ( x ) = W ( y 1 ,y 2 )( x ) . Solution: By Abel’s theorem, there is a constant c 1 such that W ( x ) = c 1 exp Z - 2 x 1 - x 2 dx = c 1 exp Z 2 x x 2 - 1 dx = c 1 exp ln ± ± x 2 - 1 ± ± = c 1 | x 2 - 1 | = c 1 (1 - x 2 ) . On the other hand, by the deﬁnition of the Wronskian, W (2) = y 1 (0) y 0 2 (0) - y 2 (0) y 0 1 (0) = 2 . Hence
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exam2 - Midterm 2 of Math 255.01(Wi11 1:30pm-2:18pm The...

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