This document provides an introduction to differential equations and their applications. It discusses the history of differential equations, types of differential equations including ordinary differential equations (ODEs) and partial differential equations (PDEs). Examples of first order ODE applications given include Newton's Law of Cooling, electrical circuits, and population growth modeling. Mechanical oscillation modeling is also discussed. The document concludes that differential equations have wide applications in fields like rocket science, economics, and gaming.

1. APPLICATION OF
DIFFERENTIAL EQUATIONS

2. BY
G.V.MANISH REDDY
II B.Sc. (M.S.Cs)
APPLICATION OF
DIFFERENTIAL EQUATIONS

3. CONTENT
 Introduction
 History of Differential Equations
 Types of Differential Equations
 Application of Differential Equations

4. INTRODUCTION
 A Differential Equation
is an equation
containing the
derivative of one or
more dependent
variables with respect
to one or more
independent variables.

5. INTRODUCTION
For Example:
𝑥
ⅆ𝑦
ⅆ𝑥
= 𝑦 − 1
ⅆ𝑦
ⅆ𝑥
= 2𝑥𝑦
𝜕𝑥
𝜕𝑦
+
𝜕𝑦
𝜕𝑥
+2=0

6.  Differential equation was part of
calculus, which itself was independently
invented by,
 English physicist Isaac Newton and
 German mathematician Gottfried
Leibniz.
History of Differential
Equations

7. History of Differential
Equations
 “Differential equations began
with Leibniz, the Bernoulli brothers, and
others from the 1680s,
 Not long after Newton’s ‘fluxional
equations’ in the 1670s.”

8. TYPES OF DIFFERENTIAL
EQUATIONS
ODE (ORDINARY DIFFERENTIAL EQUATION)
AND
TYPES OF ORDINARY DIFFERENTIAL
EQUATION
1
PDE (PARTIAL DIFFERENTIAL EQUATION)
AND
TYPES OF PARTIAL DIFFERENTIAL
EQUATION
2

9. ORDINARY
DIFFERENTIAL
EQUATION
(ODE)
 An equation contains only
ordinary derivatives of one or
more dependent variables of a
single independent variable.
 For Example,
 dy/dx + 5y = ex ,
 (dx/dt) + (dy/dt) = 2x + y

10. TYPES OF ORDINARY DIFFERENTIAL
EQUATION
I) FIRST ORDER ODE
II) SECOND ORDER ODE
III) HIGHER ORDER ODE

11. PARTIAL DIFFERENTIAL EQUATION
(PDE)
 A differential equation involving partial
derivatives of a dependent variable(one or
more) with more than one independent
variable is called a partial differential
equation, hereafter denoted as PDE.

12. TYPES OF
PARTIAL
DIFFERENTIAL
EQUATION

13. APPLICATION OF DIFFERENTIAL
EQUATIONS

14. APPLICATION OF ODE
 MODELLING WITH FIRST-ORDER EQUATIONS
 Newton’s Law of Cooling
 Electrical Circuits
 MODELLING FREE MECHANICAL OSCILLATIONS
 No Damping
 Light Damping
 Heavy Damping
 MODELLING FORCED MECHANICAL OSCILLATIONS

15. NEWTON’S
LAW OF
COOLING
 Newton’s empirical law of
cooling of an object in given by
the linear first-order differential
equation

𝑑𝑇
𝑑𝑡
= 𝛼(𝑇 − 𝑇 𝑚)
 This is a separable differential
equation. We have

𝑑𝑇
(𝑇−𝑇 𝑚)
= 𝛼𝑑𝑡
 or ln|T-Tm|=t+c1
 or T(t) = Tm+c2et

16. NEWTON’S
LAW OF
COOLING
 When a chicken is removed from
an oven, its temperature is
measured at 3000F. Three
minutes later its temperature is
200o F. How long will it take for
the chicken to cool off to a room
temperature of 70oF?

17. NEWTON’S
LAW OF
COOLING
 Solution: In (4.1) we put Tm = 70 and
T=300 at for t=0.
 T(0)=300=70+c2e.0
 This gives c2=230
 For t=3, T(3)=200
 Now we put t=3, T(3)=200 and c2=230
in (4.1) then
 200=70 + 230e.3
 Or 𝑒3𝛼 =
130
230
 Or 3𝛼 = 𝐼𝑛
13
23
 𝛼 =
1
3
In
13
23
= −0.19018

18. NEWTON’S
LAW OF
COOLING
 Thus T(t)=70+230 e-0.19018t
(4.2)
 We observe that (4.2) furnishes
no finite solution to T(t)=70
since
 limit T(t) =70.
 t 
T(min) T(t)
20.1 75º
21.3 74º
22.8 73º
24.9 72º
28.6 71º
32.3 700

19. POPULATION GROWTH
 Finding population growth using differential equations.
 The equation to find is,

𝑑𝑁
𝑑𝑡
= kN, for k = (r − m).
 Note , N(t) = N0ekt = N0e(r−m)t

20. POPULATION GROWTH
 For example,
 Given the initial condition (IC) N(0) = 6 billion, determine the
size of the human population in 100 years that our model
predicts?

21. POPULATION GROWTH
 Solution:
 We have that at time t = 0, N(0) = N0 = 6 billion.
 Then in billions, N(t)=6e0.0125t
 so that when t = 100 we would have N(100) = 6e0.0125·100 =
6e1.25 = 6 · 3.49 = 20.94
 Thus, with population around the 6 billion now, we should see
about 21 billion people on Earth in 100 years based on the
uncontrolled continuous growth model discussed here.

22. APPLICATION
OF
PDE

23. CONCLUSION
 It is also used in many fields like,
 Rocket science
 Economics
 It is useful in gaming also.

24. - MANISH