# 5. Verify the following order relations: a t2 tanht = O(t2) as t ...

## Transcribed Text

5. Verify the following order relations: a t2 tanht = O(t2) as t 80. of / ( ( ( ) - €) = O(EVE) as e 0+. f) exp(e) - 1 = O(e) as E 0. i) In € III O(E-P) as E 0+ for all p &gt; 0. or (it + =Et. 8. Use the Poincaré-Lindstedt method to obtain a two-term perturbation ap- proximation to the following problems: a) y" + y = €yy(2, y(0) = -1,y'(0)=0. = = Ey(1 - y'2), y(0) = 1,y' (0) = 0. 14. Consider the boundary value problem y" = 3ycos TTX 2 , y(0) 1. Obtain an approximation by considering the equation y"=yycos = TTX 2 for small E and then setting E = 3. How accurate is the approximatic 15 15. Consider the boundary value problem 1 Ly III t2y" + et2y' + -y = 0, O<e<< 1,<br="">4 y(1) = 1, y(e) =0. a) Use regular perturbation to find the leading-order behavior yo(t). b) Compute an upper bound for | Lyol on 1 &lt; t e when E = 0.01. Can you conclude that yo is a good approximation to the exact solution? </e<<>

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