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 )} ² s 2 Y sy (0) y ° (0) ³ + Y = 1 s 2 + 1 Use the given initial conditions and solve for Y . ² s 2 Y s · 1 2 ³ + Y = 1 s 2 + 1 ; ² s 2 + 1 ³ Y ( s + 2) = 1 s 2 + 1 ; ² s 2 + 1 ³ Y = 1 s 2 + 1 + s + 2; Y = 1 ² s 2 + 1 ³ 2 + s s 2 + 1 + 2 s 2 + 1 Invert the Laplace transform. y = L 1 ´ 1 ² s 2 + 1 ³ 2 + s s 2 + 1 + 2 s 2 + 1 µ = L 1 ´ 1 ² s 2 + 1 ³ 2 µ + L 1 s s 2 + 1 · + 2 L 1 ¸ 1 s 2 + 1 ¹ = t ) t cos( t ) 2 + t ) + 2sin( t ) = 5 2 t ) + t ) 1 2 t 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 ² s 2 Y (0) y ° (0) ³ 4 ² sY y (0) ³ + 4 Y = º 2! s 3 » s ±→ s 2 Use the given initial conditions and solve for Y . ² s 2 4 s + 4 ³ 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 ² f ( t ) ³ , where f ( t ) = ´ 2 t < 3 23 t . Rewrite in terms of U .
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hw7sol_n - [Name[Section Homework#7 MATH 527(1 Use Laplace...

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