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hw7sol_n - [Name[Section Homework#7 MATH 527(1 Use Laplace...

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[Name] [Section #] Homework #7 MATH 527 (1) “Use Laplace transforms to solve for y, where y + y = sin( t ); y (0) = 1; y (0) = 2. Apply the Laplace transform. L y + y = L {sin( t )} L y + L y = L {sin( t )} s 2 Y sy (0) y (0) + Y = 1 s 2 + 1 Use the given initial conditions and solve for Y . s 2 Y s · 1 2 + Y = 1 s 2 + 1 ; s 2 + 1 Y ( s + 2) = 1 s 2 + 1 ; s 2 + 1 Y = 1 s 2 + 1 + s + 2; Y = 1 s 2 + 1 2 + s s 2 + 1 + 2 s 2 + 1 Invert the Laplace transform. y = L 1 1 s 2 + 1 2 + s s 2 + 1 + 2 s 2 + 1 = L 1 1 s 2 + 1 2 + L 1 s s 2 + 1 + 2 L 1 1 s 2 + 1 = sin( t ) t cos( t ) 2 + cos( t ) + 2sin( t ) = 5 2 sin( t ) + cos( t ) 1 2 t cos( t ). (2) “Use Laplace transforms to solve for y, where y 4 y + 4 y = t 2 e 2 t ; y (0) = 0; y (0) = 0. Apply the Laplace transform. L y 4 y + 4 y = L t 2 e 2 t L y 4 L y + 4 L y = L t 2 s s 2 s 2 Y sy (0) y (0) 4 sY y (0) + 4 Y = 2! s 3 s s 2 Use the given initial conditions and solve for Y . s 2 4 s + 4 Y = 2 ( s 2) 3 ( s 2) 2 Y = 2 ( s 2) 3 Y = 2 ( s 2) 5 1
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2 Invert the Laplace transform. y = L 1 2 ( s 2) 5 = 1 3 · 4 L 1 4! ( s 2) 5 = 1 12 t 4 e 2 t (3) “Evaluate L f ( t ) , where f ( t ) = 2 t < 3 2 3 t . Rewrite in terms of U . f ( t ) = 2 4 U ( t 3) Apply the Laplace transform.
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