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April2003 solution

April2003 solution - 1 MATH 3705 Final Examination...

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1 MATH 3705 Final Examination Solutions April 2003 1. Lf t 3 e 2 t g =(d) (a) 6 ( s +2) 3 (b) 6 e ¡ 2 s s 3 (c) 6 ( s ¡ 2) 3 (d) 6 ( s ¡ 2) 4 (e) None of the above. 2. Lf e ¡ 3 t cos(4 t ) g =(b) (a) s ( s +3) 2 +16 (b) s +3 ( s +3) 2 +16 (c) e ¡ 3 s s 2 +16 (d) se ¡ 3 s s 2 +16 (e) None of the above. 3. L ¡ 1 ½ e ¡ 3 s s 2 ¡ 2 s +5 ¾ =(a) (a) 1 2 u ( t ¡ 3) e t ¡ 3 sin(2 t ¡ 6) (b) 1 2 u ( t ¡ 3) e t sin(2 t ¡ 6) (c) 1 2 u ( t ¡ 3) e t sin(2 t ) (d) 1 2 u ( t ¡ 3) e ¡ 3 t sin(2 t ) (e) None of the above. 4. L ¡ 1 ½ 3 s ( s 2 +9) 2 ¾ =(c)

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2 (a) t sin(3 t ) (b) ¡ t sin(3 t ) (c) 1 2 t sin(3 t ) (d) ¡ 1 2 t sin(3 t ) (e) None of the above. 5. The general solution of 4 x 2 y 00 ¡ 8 xy 0 +9 y = 0, valid for x 6 =0,isgivenby(d) (a) c 1 j x j 3 = 2 + c 2 j x j 3 = 2 (b) j x j " c 1 cos Ã p 5 2 ln j x j ! + c 2 sin Ã p 5 2 ln j x j !# (c) c 1 j x j + c 2 j x j p 5 = 2 (d) c 1 j x j 3 = 2 + c 2 j x j 3 = 2 ln j x j (e) None of the above. 6. The general solution of x 2 y 00 + xy 0 +(5 x 2 ¡ 9) y =0near x 0 = 0, valid for x> 0, is given by (b) (a) c 1 J 3 ( p 5 x )+ c 2 J ¡ 3 ( p 5 x ) (b) c 1 J 3 ( p 5 x )+ c 2 Y 3 ( p 5 x ) (c) c 1 J p 5 (3 x )+ c 2 J ¡ p 5 (3 x ) (d) c 1 J p 5 (3 x )+ c 2 Y p 5 (3 x ) (e) None of the above. 7. At x = 999, the Fourier sine series of f ( x )= x on [0 ; 1] converges to (c) (a) 1 (b) ¡ 1 (c) 0 (d) 1 2 (e) None of the above.
3 8. The di®erential equation 4

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April2003 solution - 1 MATH 3705 Final Examination...

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