Find the solution y(t) by recognizing the differential equation as determining unlimited, limited, or logistic growth, and then finding the constant.y' = 6(300 – y)= 0y =

Given differential equation - y' = 6300-yWe can write it as - dydt = 6300-ydy300-y = 6dtIntegrating…

Answered: Find the solution y(t) by recognizing…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter11: Differential Equations

Section11.1: Solutions Of Elementary And Separable Differential Equations

Problem 59E: According to the solution in Exercise 58 of the differential equation for Newtons law of cooling,...

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Find the solution y(t) by recognizing the differential equation as determining unlimited, limited, or logistic growth, and then finding the constant.
y' = 6(300 – y)
= 0
y =

Expert Solution

Step 1

$G i v e n d i f f e r e n t i a l e q u a t i o n - y' = 6 (300 - y) W e c a n w r i t e i t a s - \frac{d y}{d t} = 6 (300 - y) \frac{d y}{300 - y} = 6 d t I n t e g r a t i n g o n b o t h s i d e s - \int \frac{d y}{(300 - y)} = 6 \int d t \frac{\ln (300 - y)}{(- 1)} = 6 (t) + C [\int \frac{d x}{a x + b} = \frac{\ln (a x + b)}{a} + C] - \ln (300 - y) = 6 t + C \ln (300 - y) = - (6 t + C) . . . . . . . . . . . . . . . . . . . (1) G i v e n c o n d i t i o n - y (0) = 0 U \sin g t h i s i n e q u a t i o n a b o v e t o f i n d t h e c o n s \tan t C - \ln (300 - 0) = - [6 (0) + C] \ln (300) = - C C = - \ln (300) S u b s t i t u t i n g t h e v a l u e o f C i n e q u a t i o n (1) - \ln (300 - y) = - [6 t - \ln (300)] 300 - y = e^{- [6 t - \ln (300)]} y = 300 - e^{- [6 t - \ln (300)]}$

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