Assignment7-solution-public

Assignment7-solution-public - Math 201 Assignment #7...

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Unformatted text preview: Math 201 Assignment #7 Solutions November 18, 2011 1. Use the Laplace transform to solve y 00 + 4 y = U ( t- 2 ) sin t , y (0) = 1 , y (0) = 0 , where U ( t ) is the unit step function. Take Laplace transforms on both sides, where Y = L{ y }- y (0)- sy (0)+ s 2 Y +4 Y = L{ U ( t- 2 ) sin t } = L{ U ( t- 2 ) sin( t- 2 ) } = e- 2 s 1 s 2 + 1 Substitute in y (0) = 1 and y (0) = 0 , and isolate Y Y = s s 2 + 4 + e- 2 s 1 ( s 2 + 4)( s 2 + 1) = s s 2 + 2 2 + e- 2 s 1 / 3 s 2 + 1- 1 / 3 s 2 + 4 ! Note that the fractions in the parenthesis is derived from partial fraction. Take inverse Laplace transforms on both sides, y = L- 1 { s s 2 + 2 2 } + 1 3 L- 1 { e- 2 s 1 s 2 + 1 } - 1 6 L- 1 { 2 s 2 + 2 2 } = cos 2 t + 1 3 sin( t- 2 ) U ( t- 2 )- 1 6 sin 2( t- 2 ) U ( t- 2 ) = cos 2 t + 1 3 sin t U ( t- 2 )- 1 6 sin 2 t U ( t- 2 ) 2. Given that y = x is a solution of ( x 2 +1) y 00- 2 xy +2 y = 0 , find another linearly independent solution....
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Assignment7-solution-public - Math 201 Assignment #7...

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