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ps10sol

# ps10sol - 18.03 Problem Set 10 Fall 2009 Solutions 1(10...

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18.03 Problem Set 10 Fall 2009 Solutions 1. (10 points) This problem refers to the system considered in Problem Set 9: S ( t ) = number of students at time t, W ( t ) = number of workers at time t, P ( t ) = number of professors at time t. dS dt = - 1 4 S ( t ) + 1 48 W ( t ) dW dt = 6 25 S ( t ) - 1 50 W ( t ) dP dt = 1 100 S ( t ) - 1 50 P ( t ) . ( System of ODE ) Again we write B for the 3 × 3 matrix so that if w ( t ) = S ( t ) W ( t ) P ( t ) , then the di ff erential equation is w = B w . a) (5 points) Find a fundamental matrix X ( t ) for B with the property that the columns of X (0) are eigenvectors of B . From our solutions to Problem Set 9, we are given B = - 1 / 4 1 / 48 0 6 / 25 - 1 / 50 0 1 / 100 0 - 1 / 50 , and we have eigenvectors x 1 y 1 z 1 = - 25 / 24 1 1 / 24 , x 2 y 2 z 2 = 1 / 12 1 1 / 24 , and x 3 y 3 z 3 = 0 0 1 , with respective eigenvalues λ 1 = - 27 / 100 , λ 2 = 0 , and λ 3 = - 2 / 100. Therefore the matrix X ( t ) := - 25 24 e - 27 100 t 1 12 0 e - 27 100 t 1 0 1 24 e - 27 100 t 1 24 e - 2 100 t satisfies X ( t ) = BX ( t ). Moreover, X (0) is invertible, and the columns of X (0) are the eigenvectors x 1 y 1 z 1 , x 2 y 2 z 2 , and x 3 y 3 z 3

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