Answered: If y(x) is a solution of y' = (y+1)(50…

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If y(x) is a solution of y' = (y+1)(50 – y) with y(0) = 0, what is the limit of y(x) as x approaches +o ?

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

If y(x) is a solution of y' = (y+1)(50 – y) with y(0) = 0, what is the limit of y(x) as x
approaches + ?

Expert Solution

Step 1

Given:

$y (x)$ is a solution of $y' = (y + 1) (50 - y)$ .

Also, $y (0) = 0$ .

To find the limit of $y (x)$ as $x \to + \infty$ .

As, $y' = (y + 1) (50 - y)$ .

So further solving, we have:

$\frac{d y}{d x} = (y + 1) (50 - y) \frac{d y}{d x} = - (y + 1) (y - 50) \frac{1}{(y + 1) (y - 50)} d y = - d x \frac{1}{51} (\frac{1}{(y - 50)} - \frac{1}{(y + 1)}) d y = - d x$

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