Verify the following order relations:
a t2 tanht = O(t2) as t
of / ( ( ( ) - €) = O(EVE)
f) exp(e) - 1 = O(e) as E
i) In € III O(E-P) as
0+ for all
or (it + =Et.
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
Obtain an approximation by considering the equation
for small E and then setting E = 3. How accurate is the
15. Consider the boundary value problem
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 < t e when E = 0.01. Can
you conclude that yo is a good approximation to the exact solution?
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