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# FINAL EXAM, MATH 4140, SPRING 2009 STUDENT'S NAME: PROBLEM 1: PROBLEM 2: PROBLEM 3: PROBLEM 4: PROBLEM 5: PROBLEM 6: PROBLEM 7: TOTAL: 1 2 FINAL...

FINAL EXAM, MATH 4140, SPRING 2009 STUDENT’S NAME: PROBLEM 1: PROBLEM 2: PROBLEM 3: PROBLEM 4: PROBLEM 5: PROBLEM 6: PROBLEM 7: TOTAL: 1
2 FINAL EXAM, MATH 4140, SPRING 2009 Problem 0.1. (10 points) Let f 1 , f 2 , f : [0 , ) IR be three bounded, continuous and abso- lutely Riemann integrable functions so that | f 1 ( x ) | , | f 2 ( x ) | ≤ | f ( x ) | , for every x [0 , ) . As a consequence of the ODE theory we learned, we know that the Cauchy problem for the following system of di erential equations " α 0 ( x ) β 0 ( x ) # = " 0 f 1 ( x ) f 2 ( x ) 0 # · " α ( x ) β ( x ) # (1) with the initial condition α (0) = a, β (0) = b, has a unique solution on the whole positive real axis [0 , ) . (a) Prove that both solutions α and β can be expressed as inﬁnite series of the following type α ( x ) = a + b Z x 0 f 1 ( y ) dy + a Z x 0 f 1 ( y ) Z y 0 f 2 ( z ) dzdy + b Z x 0 f 1 ( y ) Z y 0 f 2 ( z ) Z z 0 f 1 ( w ) dwdzdy + ... and β ( x ) = b + a Z x 0 f 2 ( y ) dy + b Z x 0 f 2 ( y ) Z y 0 f 1 ( z ) dzdy + a Z x 0 f 2 ( y ) Z y 0 f 1 ( z ) Z z 0 f 2 ( w ) dwdzdy + .... (b) Show that as a consequence of this, both α and β are bounded functions and moreover they satisfy the estimate | α ( x ) | , | β ( x ) | ≤ max( | a | , | b | ) exp( k f k 1 ) (2) for every x [0 , ) , where k f k 1 : = R 0 | f ( y ) | dy. (Hint: use the Picard iteration method. )
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