Please help with this differential equation Document1 - Microsoft Word (Product Activation Failed)FileHomeInsertPage LayoutReferencesMailingsReviewViewa ?W% Cut- A AA Find -AalCalibri (Body) - 11AaAaBbCcDc AaBbCcDc AaBbC AaBbCc AaB AaBbCcLE Copyae ReplaceB I UU - abe x, x²abAI NormalI No Spaci. Heading 1PasteAChange.Heading 2TitleSubtitleFormat PainterStyles - Select -ClipboardFontParagraphStylesEditing• 2. 1:.I' 2: 1 : 3:1• 4.I 5.1' 6.1'7I'8: 1 9 ' 10.L· 11: 1 ' 12.'13 · L 14: 1· 15.1A L'17:1 18A culture initially has Ng number of bacteria. Att = 1 hour the number of bacteria is measured to be ()Ng. Ifthe rate of growth is proportional to the number of bacteria present, determine the time necessary for the numberof bacteria to triple.TO .ll07:55 AM2021-10-27Page: 1 of 1Words: 0E EA E E E100% -+)

Answered: A culture initialy has N mumber of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please help with this differential equation

Document1 - Microsoft Word (Product Activation Failed)
File
Home
Insert
Page Layout
References
Mailings
Review
View
a ?
W
% Cut
- A A
A Find -
Aal
Calibri (Body) - 11
Aa
AaBbCcDc AaBbCcDc AaBbC AaBbCc AaB AaBbCcL
E Copy
ae Replace
B I U
U - abe x, x²
ab
A
I Normal
I No Spaci. Heading 1
Paste
A
Change
.
Heading 2
Title
Subtitle
Format Painter
Styles - Select -
Clipboard
Font
Paragraph
Styles
Editing
• 2. 1:.
I' 2: 1 : 3:1
• 4.I 5.1' 6.1'7
I'8: 1 9 ' 10.L· 11: 1 ' 12.'13 · L 14: 1· 15.1A L'17:1 18
A culture initially has Ng number of bacteria. Att = 1 hour the number of bacteria is measured to be ()Ng. If
the rate of growth is proportional to the number of bacteria present, determine the time necessary for the number
of bacteria to triple.
TO .ll
07:55 AM
2021-10-27
Page: 1 of 1
Words: 0
E EA E E E
100% -
+)

Expert Solution

Step 1

Let N be the number of bacteria at time t.

The rate of increase which is proportional to N = $\frac{d N}{d t}$

Given

$\frac{d N}{d t} \propto N$

$\frac{d N}{d t} = k N$ , where k is any constant.

$\int \frac{d N}{N} = k \int d t + c$

$\log N = k t + C A t t = 0, N = N_{0} \Rightarrow \log N_{0} = C \Rightarrow \log N = k t + \log N_{0} A t t = 1, N = \frac{3}{2} N_{0} \log (\frac{3}{2} N_{0}) = k + \log N_{0} k = \log (\frac{3}{2}) T h e r e f o r e, \log N = t (\log (\frac{3}{2})) + \log N_{0} i f N = 3 N_{0} \Rightarrow \log 3 N_{0} = t (\log (\frac{3}{2})) + \log N_{0} \Rightarrow \log (3) = t (\log \frac{3}{2}) \Rightarrow t = 2.7$