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T1Solutions - MATH 3705C Test 1 Solutions Questions 1-4 are...

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MATH 3705C Test 1 Solutions January 26, 2007 Questions 1-4 are multiple choice. Circle the correct answer. Only the answer will be marked. [Marks] 1. Lf e ¡ 2 t cos(3 t ) g = [2] (a) s ( s ¡ 2) 2 + 9 (b) s ¡ 2 s 2 + 9 (c) s ¡ 2 ( s ¡ 2) 2 + 9 (d) s + 2 ( s + 2) 2 + 9 (e) None of these Solution: (d) 2. Lf u ( t ¡ 3) e ¡ 2 t g = [2] (a) e ¡ 3( s +2) s + 2 (b) e ¡ 3 s s + 2 (c) e 6 e ¡ 3 s s + 2 (d) e 3 s s + 2 (e) None of these Solution: (a) 3. L ¡ 1 f se ¡ s s 2 ¡ 2 s + 4 g = [3] (a) e t cos( p 3 t ) + 1 3 e 3 t sin( p 3 t ) (b) u ( t ¡ 1) e t ¡ 1 © cos £ p 3( t ¡ 1) ¤ + sin £ p 3( t ¡ 1) ¤ª (c) u ( t ¡ 1) e t ¡ 1 ½ cos h p 3( t ¡ 1) i + 1 p 3 sin h p 3( t ¡ 1) i ¾ (d) u ( t ¡ 1) e t ¡ 1 ½ cos h p 3( t ¡ 1) i + 1 3 sin h p 3( t ¡ 1) i ¾ (e) None of the above Solution: (c) 4. L ¡ 1 f 6 s ( s 2 + 9) 2 g = [3] (a) t sin(3 t ) (b) 2 t sin(3 t ) (c) ¡ t sin(3 t ) (d) ¡ 2 t sin(3 t ) (e) None of these Solution: (a) 5. Solve the initial-value problem y 00 + y 0 ¡ 6 y = 0 ; y (0) = 0 ; y 0 (0) = 1. [6] Solution: [ s 2 Y ( s ) ¡ sy (0) ¡ y 0 (0)] + [ sY ( s ) ¡ y (0)] ¡ 6 Y ( s ) = 0 ) ( s 2 + s ¡ 6) Y ( s ) = 1 ) Y ( s ) = 1 s 2 + s ¡ 6 = 1 ( s + 3)( s ¡ 2) = A s + 3 + B s ¡ 2 = A ( s ¡ 2) + B ( s + 3) ( s + 3)( s ¡ 2) ) A + B = 0 ; ¡ 2 A + 3 B = 1 ) A = ¡ 1 5 ; B = 1 5 ) Y ( s ) = 1 5 1 s ¡ 2 ¡ 1 5 1 s + 3 ) y ( t ) = 1 5 e 2 t ¡ 1 5 e ¡ 3 t .
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2 6. Solve the initial-value problem y 00 + 4 y 0 + 8 y = 0 ; y (0) = 2 ; y 0 (0) = 4. [6] Solution: [ s 2 Y ( s ) ¡ sy (0) ¡ y 0 (0)] + 4[ sY ( s ) ¡ y (0)] + 8 Y ( s ) = 0 ) ( s 2 + 4 s + 8) Y ( s ) ¡ 2 s ¡ 12 = 0 ) Y ( s ) = 2 s + 6 s 2 + 4 s + 8 = 2 ( s + 2) + 4 ( s + 2) 2 + 4 = 2
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