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.

7th Edition

ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 2.3, Problem 37E

To determine

To find:

The value of x(t).

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Chapter 2.3, Problem 36E

Chapter 2.3, Problem 38E

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