midterm3-B-solution-public

midterm3-B-solution-public - Math 201 Fall 2011 Midterm...

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Math 201 Fall 2011 Midterm Exam 3 (B) November 29, 2011 1. [5] Set up (but do not solve) the diﬀerential equation of motion for a spring-mass system where the mass is 2 units, the damping coeﬃcient is 2 and the spring constant is 2 with an external applied force F ( t ) = 0 , 0 t < 1 , 2 , 1 t < 2 , 0 , t > 2 , using the unit step function; The equation is mx 00 + γx 0 + kx = F ( t ) , where the mass m = 2 , the spring constant k = 2 , the damping coeﬃcient γ = 2 . F ( t ) = 2[ U ( t - 1) - U ( t - 2)] . So, the equation is 2 x 00 + 2 x 0 + 2 x = 2 U ( t - 1) - 2 U ( t - 2) or, simplify it, x 00 + x 0 + x = U ( t - 1) - U ( t - 2) 2. [5] Find the general solution of x 2 y 00 + 3 xy 0 + 1 y = 0 . The auxiliary equation is m ( m - 1)+3 m +1 = m 2 +2 m +1 = ( m +1) 2 = 0 , a repeated root m = - 1 . Thus the two linearly independent solutions are y 1 = x - 1 and y 2 = y 1 ln x = x - 1 ln x . The general solution is thus

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midterm3-B-solution-public - Math 201 Fall 2011 Midterm...

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