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202s05fin - B U Department of Mathematics Math 202...

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B U Department of Mathematics Math 202 Differential Equations Spring 2005 Final Exam This archive is a property of Bo˘ gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. Let L ( f ( t ) ) = F ( s ). Show that L ( f ( at ) ) = 1 a F ( s a ) , a > 0 . Solution: L ( f ( at ) ) = 0 e - st f ( at ) dt = 0 1 a e - su a f ( u ) du ( u = at ; du = adt ) = 1 a F ( s a ) . 2. Solve the following initial value problem and discuss the interval of existence (1 + t ) x + x = cos t ; x ( - π 2 ) = 0 . Solution: Observe that the left hand side equals d dt ((1 + t ) x ). Then (1 + t ) x = cos t dt = sin t + C, ( C R ) . Inserting the initial condition we get: 0 = sin( - π 2 ) + C. Hence C = 1 and x ( t ) = sin t + 1 1 + t . 3. Given that y ( t ) = e - t sin t is a solution of the constant-coefficient differential equation 9 y + 11 y + 4 y - 14 y = 0 , find the general solution of this equation.
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