Explanation Given: Population growth model for population changed either by immigrations d y d x = k y + f ( t ) , k = 0.02 y ( 0 ) = 10000 f ( t ) = − t Approach: General form of linear differential equation of first order is d y d x + P ( x ) y = Q ( x ) Steps to solve linear differential equation of first order are given below: Write given equation in standard form of first order differential equation. Find value of P ( x ) and Q ( x ) . Find integrating factor as I . F = e ∫ P ( x ) d x . Solution of first order linear differential equation is in the form as given below y × ( I . F ) = ∫ ( I . F ) × Q ( x ) d x + C Use initial condition to solve for value of C . Calculation: Given equation is d y d t = k y + f ( t ) and k = 0.02 y ( 0 ) = 10000 f ( t ) = − t We can rearrange equation as: d y d t = 0.02 y − t d y d t − 0.02 y = − t On comparing with d y d t + P ( t ) y = Q ( t ) P ( t ) = − 0.02 , Q ( t ) = − t Hence I . F = e ∫ − 0.02 d t = e − 0

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

2nd Edition

ISBN: 9780137553457

Author: Raymond Greenwell, Nathan Ritchey

Publisher: PEARSON+

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Chapter 11.2, Problem 31E

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The differential equation dydx=ky+f(t) which is subjected to initial condition y(0)=10000 for the function f(t)=−t.

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Chapter 11.2, Problem 30E

Chapter 11.2, Problem 32E

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Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

Ch. 11.1 - YOUR TURN 1 Find all solutions of the differential...Ch. 11.1 - Prob. 2YT Ch. 11.1 - Prob. 3YT Ch. 11.1 - YOUR TURN In Example 6, find the goat population...Ch. 11.1 - Find the general solution for each differential...Ch. 11.1 - Prob. 2E Ch. 11.1 - Find the general solution for each differential...Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E

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The differential equation d y d x = k y + f ( t ) which is subjected to initial condition y ( 0 ) = 10000 for the function f ( t ) = − t .

Solution Summary: The author explains how to solve a linear differential equation of first order.

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To find:

The differential equation dydx=ky+f(t) which is subjected to initial condition y(0)=10000 for the function f(t)=−t.

Want to see the full answer?

Chapter 11 Solutions

The differential equation d y d x = k y + f ( t ) which is subjected to initial condition y ( 0 ) = 10000 for the function f ( t ) = − t .

Solution Summary: The author explains how to solve a linear differential equation of first order.

Concept explainers

Videos

To find: The differential equation dydx=ky+f(t) which is subjected to initial condition y(0)=10000 for the function f(t)=−t.

Want to see the full answer?

Chapter 11 Solutions

To find:

The differential equation dydx=ky+f(t) which is subjected to initial condition y(0)=10000 for the function f(t)=−t.