April2004 solution

# April2004 solution - MATH 3705 Final Examination Solutions...

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MATH 3705 Final Examination Solutions April 2004 1. Lf e 2 t cos(3 t ) g =(a) (a) s ¡ 2 ( s ¡ 2) 2 +9 (b) s ( s ¡ 2) 2 +9 (c) s ¡ 2 s 2 +9 (d) e ¡ 2 s s 2 +9 (e) None of the above. 2. Lf t sin(2 t ) g =(d) (a) s ( s 2 +4) 2 (b) ¡ s ( s 2 +4) 2 (c) 2 ( s ¡ 1) 2 +4 (d) 4 s ( s 2 +4) 2 (e) None of the above. 3. L ¡ 1 ½ 3 e ¡ 2 s s 2 + s ¡ 2 ¾ =(b) (a) u ( t ¡ 2) [ e t ¡ e ¡ 2 t ] (b) u ( t ¡ 2) [ e t ¡ 2 ¡ e ¡ 2 t +4 ] (c) e t ¡ 2 ¡ e ¡ 2( t ¡ 2) (d) u ( t ) e t ¡ u ( t ¡ 2) e t ¡ 2 (e) None of the above.

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2 4. L ¡ 1 ½ s s 2 +2 s +10 ¾ =(d) (a) e ¡ t cos(3 t ) (b) e ¡ t sin(3 t ) (c) e ¡ t cos(3 t ) ¡ e ¡ t sin(3 t ) (d) e ¡ t [cos(3 t ) ¡ 1 3 sin(3 t )] (e) None of the above. 5. The general solution of the di®erential equation 2 x 2 y 00 ¡ 5 xy 0 +3 y = 0, valid for x 6 =0 ,i s given by (a) (a) c 1 j x j 3 + c 2 j x j 1 2 (b) j x j 3 [ c 1 + c 2 ln j x j ] (c) j x j 3 c 1 cos μ 1 2 ln j x j + c 2 sin μ 1 2 ln j x j ¶¸ (d) c 1 j x j 1 2 + c 2 j x j 3 ln j x j (e) None of the above. 6. The general solution of the di®erential equation x 2 y 00 +2 xy 0 + 1 4 y = 0, valid for x 6 =0 ,isg iven by (d) (a) j x j ¡ 1 " c 1 cos Ã p 3 2 ln j x j ! + c 2 sin Ã p 3 2 ln j x j !# (b) c 1 j x j ¡ 1 + c 2 j x j p 3 2 (c) c 1 j x j ¡ 1 2 + c 2 j x j ¡ 1 2 (d) j x j ¡ 1 2 ( c 1 + c 2 ln j
3 7. The general solution of the di®erential equation x 2 y 00 + xy 0 +(7 x 2 ¡ 4) y = 0, valid for x> 0, is given by (c) (a) c 1 J 2 ( p 7 x )+ c 2 J ¡ 2 ( p 7

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April2004 solution - MATH 3705 Final Examination Solutions...

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