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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 π ) 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 π ) sin t } = L{ U ( t 2 π ) sin( t 2 π ) } = e 2 πs 1 s 2 + 1 • Substitute in y (0) = 1 and y (0) = 0 , and isolate Y Y = s s 2 + 4 + e 2 πs 1 ( s 2 + 4)( s 2 + 1) = s s 2 + 2 2 + e 2 πs 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 πs 1 s 2 + 1 }  1 6 L 1 { 2 s 2 + 2 2 } = cos 2 t + 1 3 sin( t 2 π ) U ( t 2 π ) 1 6 sin 2( t 2 π ) U ( t 2 π ) = cos 2 t + 1 3 sin t U ( t 2 π ) 1 6 sin 2 t U ( t 2 π ) 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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 Winter '10
 STEACY
 Math, Laplace, Cxy, ln x2

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