Application Of Differential Equation PDF

1
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

2
Table of Contents
INTRODUCTION.......................................................................................................................................3
2. MOTIVATING EXAMPLES.....................................................................................................................3
Motivating example-1.............................................................................................................................3
Motivating example-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
3.2 In biology...........................................................................................................................................5
3.3 In chemistry.......................................................................................................................................5
APPLICATION OF DIFFERENTIAL EQUATION IN FALLING OBJECT.............................................................5
APPLICATION OF 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

3
INTRODUCTION
The Differential equations have wide applications in various engineering and
science disciplines. In general, modeling of the variation of a physical quantity, such as
temperature, pressure, displacement, velocity, stress, strain, current, voltage, or
concentration of a pollutant, with the change of time or location, or both would result in
differential equations. Similarly, studying the variation of some physical quantities on
other physical quantities would also lead to differential equations. In fact, many
engineering subjects, such as mechanical vibration or structural dynamics, heat
transfer, or theory of electric circuits, are founded on the theory of differential
equations. It is practically important for engineers to be able to model physical
problems using mathematical equations, and then solve these equations so that the
behavior of the systems concerned can be studied.
2. MOTIVATING EXAMPLES
It is important for engineers to be able to model physical problems using mathematical
equations, and then solve these equations so that the behavior of the systems
concerned can be studied. In this section, a few examples are presented to illustrate
how practical problems are modeled mathematically and how differential equations
arise in them.
Motivating example-1
A tank contains a liquid of volume V(t), which is polluted with a pollutant concentration
in percentage of c(t) at time t. To reduce the pollutant concentration, an inflow of rate
Qin is injected to the tank. Unfortunately, the inflow is also polluted but to a lesser
degree with a pollutant concentration cin. It is assumed that the inflow is perfectly
mixed with the liquid in the tank instantaneously. An outflow of rate Qout is removed
from the tank. Suppose that, at time t = 0, the volume of the liquid is V0 with a
pollutant concentration of c0.The equation governing the pollutant concentration c(t)
is given by
[V0 + (Qin - Qout)t] (dc(t)/dt)+ Qin c(t) = Qin cin;
With initial condition c(0) = c0. This is a first-order ordinary differential equation
Motivating example-2
Consider the suspension bridge, which consists of the main cable, the hangers, and
the deck. The self-weight of the deck and the loads applied on the deck are
transferred to the cable through the hangers. Set up the Cartesian coordinate system