m238sum08t4 - (your answer should have two constants of...

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name : Mathematics 238 test four Wednesday, August 13, 2008 please show your work to get full credit for each problem 1 . Sketch a graph of the function f ( t ) = (1 - u 2 ( t )) (2 t ) + ( u 2 ( t ) - u 4 ( t )) (6 - t ) + u 5 ( t ) 2 . Calculate the Laplace transform of the f ( t ) = 4 - 2 t + 5 e - 2 t sin 3 t + 8 δ ( t ) - ( u 3 ( t ) - u 5 ( t )) ( t - 3) 3 . Evaluate each integral: ( HINT : think about what each integral represents) ( a ) Z 0 ± t 5 - u 4 ( t ) ² e - st dt ( b ) Z t 0 s 3 ( t - s ) 4 ds 4 . Simplify: ( a ) (cos t ) δ ( t - π ) ( b ) u 3 ( t ) ( u 7 ( t ) - δ ( t - 2)) ( c ) the convolution e t * e - t 5 . Find the inverse Laplace transform of ( a ) 3 s + 4 ( s - 2)( s + 3) ( b ) 3 s - 2 s 2 + 4 s + 13 ( c ) 4 ( s + 1) 2 - e - 3 s ( s + 1) 2 6 . Use the Laplace transform to solve each initial value problem: ( a ) y 00 +6 y 0 +9 y = 6 te - 3 t and y (0) = 0 and y 0 (0) = 1 ( b ) y 00 +9 y = u 2 ( t ) - u 4 ( t ) and y (0) = 0 and y 0 (0) = 0 7 . Find the general solution to the system of differential equations:
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Unformatted text preview: (your answer should have two constants of integration) dx dt =-x + y dy dt = 5 x + 3 y 8 . Given the dierential equation y 00 + 2 xy = 0, nd a general solution in the format of a power series around x = 0 ( a ) for which values of x will this power series converge? ( b ) nd a recursion relation for the coecients of a power series solution about x = 0 ( c ) write down the general solution through the degree ve term....
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This note was uploaded on 11/01/2010 for the course ENGINEERIN PHYS222 taught by Professor Ewqfqw during the Spring '09 term at University of Washington.

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