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Unformatted text preview: < c 2 is sufciently small, then their is a continuous function u ( x ) that satises u ( x ) = f ( x )+ Z c h ( x , y ) u ( y ) dy . (*) b) In the special case where h ( x , y ) 1 and f ( x ) 1, solve equation (*) explicitly. [This is easy. Let = R 1 u ( y ) dy and then use (*) to solve for ]. From this, show that indeed for some value of c a solution may not exist. 6. Let f ( x ) and h ( x , y ) be as in the previous problem. Show that if > 0 is sufciently small, the equation u ( x ) = f ( x )+ Z 2 h ( x , y ) u ( y ) dy . (1) has a unique continuous solution u ( x ) . 7. Let f be an even continuous function on [1 , 1 ] with R 11 f ( x ) x n dx = 0 for all even n 0. Show that f 0. [Last revised: December 8, 2010] 1...
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.
 Fall '10
 STAFF
 Derivative

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