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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Writing and Solving a Differential Equation In Exercises 21 and 22, write and find the general solution of the differential equation that models the verbal statement. Evaluate the solution at the specified value of the independent variable.

The rate of change of N is proportional to N. When t = 0 , N = 250 , and when t = 1 , N = 400 . What is the value of N when t = 4 ?

To determine

To calculate: General solution for the differential equation that models the verbal statement, ‘The rate of change of N is proportional to N’ and finding the solution for N at t=4.

Explanation

Given:

The change in rate of N is proportional to N.

For t=0 the value of N=250 and also for t=1 the value of N=400.

Formula used:

Integration of xn is given by:

xndx=xn+1n+1+C

Integration of 1x is given as,

1xdx=log|x|+C

Calculation:

As change in rate of N is proportional to N, we have:

dNdtN

By using proportionality constant k in above equation:

dNdt=kN

By multiplying both sides by dt, we get:

dNdt×dt=kNdtdN=kΝdtdNN=kdt

Next, by integrating on both sides:

dNN =kdtln|N| =kt+C1eln|N|=ekt+c1N=ekt+c1

By simplifying further,

N=ekt+c1N=ekt×ec1

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