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m238sp05t2 - Dierential Equations test two Monday please...

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Differential Equations test two Monday, May 16, 2005 please show your work in order to get full credit 1 . For which values of t is the solution to the following differential equation guaranteed to exist? ( t 2 - 9) y + (cos t ) y + y = 2 t and y (0) = - 4 2 . Calculate the Wronskian of the functions f ( t ) = e 2 t and g ( t ) = e - t . 3 . Which of the following are linearly independent sets of functions? ( a ) { f ( t ) = t, g ( t ) = t 2 } ( b ) { f ( t ) = cos t, g ( t ) = t cos t } ( c ) { f ( t ) = e t - sin t, g ( t ) = sin t - e t } 4 . Find the real part of the complex-valued function f ( t ) = (2 + 3 i ) e ( - 1+4 i ) t . 5 . Find the real-valued general solution of each differential equation: ( a ) y - 2 y - 8 y = 0 ( b ) y + 4 y + 40 y = 0 ( c ) D ( D - 2)( D + 4) 2 ( D 2 + 4) y = 0 ( d ) y + 2 y = 12 - 4 cos 2 t 6 . Solve the initial value problem: y - 2 y - 8 y = 0 and y (0) = 1 and y (0) = 0. 7 . Given the differential equation y - 2 y - 8 y = 9 + 18 e t find a particular solution of this differential equation. find the general solution of this equation. find a solution which satisfies initial conditions
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