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m251_problem_set2[fa10]

m251_problem_set2[fa10] - MATH 251/12 Problem set#2 Due 1...

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MATH 251.6/8/11/12 Problem set #2 Due: 11-11-2010 1. Find the general solution of (a) y (6) - 5 y (4) - 36 y ″ = 0 (b) y (6) - 2 y (4) + y ″ = 0 2. Find the Laplace transform of (a) u π/4 ( t ) e 3 t cos t . (b) u 2 ( t ) ( t 3 t + 4) 3. Find the Laplace transform of (a) e -5 t sin(4 t ) cos(4 t ). (b) 6sin(8 t ) sin(2 t ) 4. Find the inverse Laplace transform of each given function: (a) F ( s ) = e - s (7 - 3 s ) / ( s 2 - 8 s + 20) (b) F ( s ) = ( s 2 + 5) / ( s - 2) 3 (c) F ( s ) = 5 e -3 s s / ( s 4 - 16) (d) F ( s ) = e s / ( s 4 + 16 s 2 ) 5. Use the method of Laplace transform to solve the following 1st order linear IVP. y ′ + 6 y = e -5 t + u 8 ( t ), y (0) = 3 6. Use the method of Laplace transform to solve the following 3rd order linear IVP. y ″′ - y ″ - 2 y ′ = 0, y (0) = 0, y ′(0) = 1, y ″(0) = 5 7. Find the general solution for each system. Then classify the type and stability of the critical point at (0,0). (a) x ′ = - 3 0 4 2 x . (b) x ′ = 4 2 0 4 x . (c) x ′ = - 1 1 4 1 x . (d) x ′ = - - 6 8 5 6 x .
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