Explanation Given: The rate of change of the level of hormone in the blood is given by, d x d t = α − β cos π t 12 − k x , x ( 0 ) = x 0 here x ( t ) is the amount of the hormone in the blood at time t , α is the average secretion rate, β is the amount of daily variation in the secretion, and k is a positive constant reflecting the rate at which the body removes the hormone from the blood. And α = 1 , β = 1 , k = 2 , and x 0 = 10 . Formula used: The first order linear differential equation is given by, x ′ + p ( t ) x = q ( t ) And the integrating factor I ( t ) for the above equation is given as follows. I ( t ) = e ∫ p ( t ) d t Calculation: Consider the differential equation. d x d t = α − β cos π t 12 − k x Substitute 1 for α , 1 for β and 2 for k in above equation. d x d t = 1 − cos π t 12 − 2 x d x d t + 2 x = 1 − cos π t 12 ... ( 1 ) Compare the expression with x ′ + p ( t ) x = q ( t ) . This gives, p ( t ) = 2 q ( t ) = 1 − cos π t 12 Consider integrating factor. I ( t ) = e ∫ p ( t ) d t Substitute 2 for p ( t ) in above equation. I ( t ) = e ∫ 2 d t = e 2 t Multiply by e 2 t in equation ( 1 ) . e 2 t ( d x d t + 2 x ) = e 2 t ( 1 − cos π t 12 ) d d t [ e 2 t x ( t ) ] = e 2 t − e 2 t cos π t 12 Integrate the both side of above equation.

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

7th Edition

ISBN: 9780137394524

Author: R. Nagle, Edward Saff

Publisher: PEARSON+

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