Explanation Given: A model for the spread of an infectious disease among mice is given by d N d t = r N − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] where N is the size of the population of mice, α is the mortality rate due to infection, b is the mortality rate due to natural causes, β is the transmission coefficient for the rate that infected mice infect susceptible mice, v is the rate mice recover from infection and γ is the rate that mice lose immunity. Calculation: It is given that the equation is d N d t = r N − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] . Rewrite the given equation as follows: d N d t − r N = − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] ...... ( 1 ) Now, it is observed that the above equation is in the form of first order differential equation. Therefore, the integrating factor is calculated as follows: I = e ∫ − r d t = e − r t Multiply the integrating factor I = e − r t in equation (1) as follows: e − r t ( d N d t − r N ) = e − r t ( − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] ) e − r t ⋅ d N d t − e − r t ⋅ ( − r N ) = e − r t ( − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] ) D t ( N e − r t ) = − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] e − r t Integrate the above equation. ∫ D t ( N e − r t ) = ∫ − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] e − r t N e − r t = − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] ∫ e − r t N e − r t = − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] [ − 1 r e − r t ] + C N ( t ) = α ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] + C e r t The obtained equation is given below

Calculus with Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Lial, Greenwell & Ritchey, The Applied Calculus & Finite Math Series)

11th Edition

ISBN: 9780133886832

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Differential Equations

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Chapter 10.2, Problem 26E

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To show: The solution of the equation dNdt=rN−αr(α+b+v)β[α−r(1+vb+γ)] for the spread of an infectious disease among mice is N(t)=(α+b+ν)βR[(R−α)ert+α] where R=α −r(1+νb+γ).

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Chapter 10.2, Problem 25E

Chapter 10.2, Problem 27E

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Calculus with Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Lial, Greenwell & Ritchey, The Applied Calculus & Finite Math Series)

Ch. 10.1 - Find all solutions of the differential equation . Ch. 10.1 - Prob. 2YT Ch. 10.1 - Prob. 3YT Ch. 10.1 - Prob. 4YT Ch. 10.1 - Prob. 1WE Ch. 10.1 - Prob. 2WE Ch. 10.1 - Prob. 3WE Ch. 10.1 - Prob. 4WE Ch. 10.1 - Prob. 5WE Ch. 10.1 - Find the general solution for each differential...

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The solution of the equation d N d t = r N − α r ( α + b + v ) β [ α − r ( 1 + v b + γ ) ] for the spread of an infectious disease among mice is N ( t ) = ( α + b + ν ) β R [ ( R − α ) e r t + α ] where R = α − r ( 1 + ν b + γ ) .

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To show: The solution of the equation dNdt=rN−αr(α+b+v)β[α−r(1+vb+γ)] for the spread of an infectious disease among mice is N(t)=(α+b+ν)βR[(R−α)ert+α] where R=α −r(1+νb+γ).

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