AMS 501 Sample Questions for Final Exam

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AMS 501 Sample Questions for Final Exam. December 8, 2011. 1. (20 points) Find eAt, where A = ⎡. ⎣. 1 2 3. 0 1 3. 0 0 1. ⎤. ⎦ . 2. (20 points) For the ...
AMS 501 Sample Questions for Final Exam December 8, 2011 

1 1. (20 points) Find eAt , where A =  0 0

2 1 0

 3 3 . 1

2. (20 points) For the Sturm-Liouville problem y 00 + λy

=

0

y(0)

=

0,

(0 < x < L) hy(L) − y 0 (L) = 0,

show that it has a single negative eigenvalue λ0 if and only if hL > 1, in which case λ0 = −β02 /L2 and its associated eigenfunction is y0 (x) = sinh (β0 x/L), where β0 is the positive root of the equation tanh x = x/hL. 3. (20 points) Find two linearly independent Frobenius solutions, or find one such solution and show that a second such solution does not exist: x2 y 00 + (2x + 3x2 )y 0 − 2y = 0. 4. (20 points) Consider the differential equation x2 y 00 + 3xy 0 + (1 + x2 )y = 0. (a) Convert it into a Bessel equation. (b) Express its general solution in terms of Bessel functions. 5. (20 points) Find the leading behavior of the following equation as x → 0+: x5 y 000 − 2xy 0 + y = 0.

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