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Application Of Differential Equation PDF

Added on - 19 Jan 2022

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Assignment no #1
Application Of Differential Equation
Name:Nauman Tariq
Roll No:19011519-088
Section:BSCS-19-IV-A
Course:Differential Equation
Course Code:CS-
Submission Date:12thjan 2022
Submission To:Dr. Aniqa Zulfiqar
DEPARTMENT OF COMPUTER SCIENCE
UNIVERSITY OF GUJRAT
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Table of Contents
INTRODUCTION.......................................................................................................................................3
2. MOTIVATINGEXAMPLES.....................................................................................................................3
Motivatingexample-1.............................................................................................................................3
Motivatingexample-2.............................................................................................................................3
Motivating example-3.............................................................................................................................3
APPLICATIONS AND CONNECTIONS TO OTHER AREAS............................................................................4
3.1 In physics...........................................................................................................................................4
1) Classical mechanics:...........................................................................................................................4
2) Electrodynamics:.................................................................................................................................4
3) General relativity:................................................................................................................................4
4) Quantum mechanics:...........................................................................................................................4
5) Other important equations:................................................................................................................5
3.2In biology...........................................................................................................................................5
3.3 In chemistry.......................................................................................................................................5
APPLICATION OF DIFFERENTIAL EQUATION IN FALLING OBJECT.............................................................5
APPLICATIONOF DIFFERENTIAL EQUATION IN NEWTON’S LAW OF COOLING...................................6
5.3 Radio-active Decay and Carbon Dating..............................................................................................7
5.4 Carbon Dating....................................................................................................................................7
6.CONCLUSION........................................................................................................................................8
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INTRODUCTION
The Differential equations have wide applications in various engineering and
science disciplines. In general, modeling ofthe variationofa physical quantity, such as
temperature,pressure,displacement,velocity,stress,strain,current,voltage,or
concentrationofa pollutant, with the changeoftimeorlocation,or both wouldresult in
differential equations. Similarly, studying the variationofsome physical quantitieson
otherphysicalquantitieswouldalsoleadtodifferentialequations.Infact,many
engineeringsubjects,suchasmechanicalvibrationorstructuraldynamics,heat
transfer,ortheoryofelectriccircuits,arefoundedonthetheoryofdifferential
equations.Itispracticallyimportantforengineerstobeabletomodelphysical
problems using mathematical equations, and thensolvethese equations so that the
behaviorofthesystemsconcerned canbestudied.
2. MOTIVATINGEXAMPLES
It is importantforengineers tobeable to model physical problems using mathematical
equations,andthensolvethese equations so that the behavior ofthesystems
concerned canbestudied. In this section, afewexamples are presented to illustrate
how practical problems are modeled mathematically and how differential equations
arise inthem.
Motivatingexample-1
A tank contains a liquidofvolume V(t), which is polluted with a pollutant concentration
in percentageofc(t) at time t. To reduce the pollutant concentration, an inflowofrate
Qinis injected to the tank. Unfortunately, the inflow is also polluted but to a lesser
degree with a pollutant concentrationcin. It is assumed that the inflow is perfectly
mixed withthe liquid in the tank instantaneously. An outflowofrate Qoutis removed
from the tank. Suppose that, at time t = 0, the volumeofthe liquid is V0with a
pollutant concentrationofc0.The equationgoverning the pollutant concentration c(t)
is givenby
[V0+ (Qin- Qout)t] (dc(t)/dt)+ Qinc(t) = Qincin;
With initial condition c(0) = c0. This is a first-order ordinary differential equation
Motivatingexample-2
Consider the suspension bridge, which consistsofthe main cable, the hangers, and
thedeck.Theself-weightofthedeckandtheloadsappliedonthedeckare
transferred to the cable through the hangers. Set up the Cartesian coordinatesystem
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