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HW AnswerGuide6 - Answer Guide 6 Math 427K Unique Number...

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Answer Guide 6 Math 427K: Unique Number 55160 Wednesday, October 12, 2011 1. Suppose that x ( t ) = cos( ! 1 t ) ° cos( ! 2 t ) . Make the substitutions ° = ! 1 + ! 2 2 and ± = ! 1 ° ! 2 2 and verify that x ( t ) = ° 2 sin( °t ) sin( ±t ) : Solution: Following the hint, let A = °t = ! 1 + ! 2 2 t and B = ±t = ! 1 ° ! 2 2 t . Then A + B = ! 1 t and A ° B = ! 2 t . Using the angle addition formulas, we easily get x ( t ) = cos( A + B ) ° cos( A ° B ) (1a) = f cos A cos B ° sin A sin B g ° f cos A cos B + sin A sin B g (1b) = ° 2 sin A sin B (1c) = ° 2 sin( °t ) sin( ±t ) : (1d) For the next four problems, °nd the general solution: 2. x 000 ° x 00 ° 12 x 0 = 0 : Solution: The characteristic equation is 0 = r 3 ° r 2 ° 12 r = r ( r + 3)( r ° 4) : So the general solution is x ( t ) = C 1 e ° 3 t + C 2 + C 2 e 4 t : ( F ) 3. x 0000 ° 4 x 000 + 7 x 00 ° 16 x 0 + 12 x = 0 : Solution: The characteristic equation is 0 = r 4 ° 4 r 3 + 7 r 2 ° 16 r + 12 = ( r ° 1)( r ° 3)( r ° 2 i )( r + 2 i ) : So the general solution is x ( t ) = C 1 e t + C 2 e 3 t + C 3 cos(2 t ) + C 4 sin(2 t ) : ( F )
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4. x 000 + x = 0 : Solution: The characteristic equation is 0 = r 3 + 1 = ( r + 1)( r 2 ° r + 1) :
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