MYLAB MATH-W/ETEXT F/FUND.DIFF.EQUAT.
MYLAB MATH-W/ETEXT F/FUND.DIFF.EQUAT.
7th Edition
ISBN: 9780135902738
Author: Nagle
Publisher: PEARSON
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Chapter 13.4, Problem 11E

Let f ( x , y ) = y 2 .

  1. Solve explicitly for ϕ ( x , y ) , the solution to

    y = f ( x , y ) , y ( 0 ) = 1 and for ϕ ( x , 1 + ε ) , the solution to

    y = f ( x , y ) , y ( 0 ) = 1 + ε , where ε > 0 .

  2. Show that | f / y | 4 on the rectangle

    R = { ( x , y ) : 1 < x < 1 , 2 < y < 2 } .

  3. Using the result of part (b) in inequality ( 8 ) in Theorem 9 , it appears that

    | ϕ ( x , 1 ) ϕ ( x , 1 + ε ) | ε e 4 , x in ( 1 , 1 ) .

    Show that this is not true (even for ε > 0 small) by using the results of part(a) to verify that ϕ ( x , 1 ) 1 + ε 1 as x 1 1 + ε , while | ϕ ( x , 1 + ε ) | + as x 1 1 + ε .

  4. Which assumption in Theorem 9 is not satisfied so that inequality ( 8 ) is not applicable to this situation?
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MYLAB MATH-W/ETEXT F/FUND.DIFF.EQUAT.

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