This document discusses the application of ordinary differential equations. It begins with a brief history of differential equations, noting they were independently invented by Newton and Leibniz. It then defines an ordinary differential equation as one that contains derivatives of dependent variables with respect to a single independent variable. The document goes on to list several examples of applying ordinary differential equations, including Newton's Law of Cooling, modeling mechanical oscillations, radioactive decay, electrical circuits, and bending beams. It provides the specific differential equation that models Newton's Law of Cooling and radioactive decay of elements.

2. Application of Ordinary
Differential Equation
Presenting to: Sir Shehmazar Baloch
Prepared By: NOOR AHMED
Roll NO: 17CE71

3. INVENTION OF DIFFERENTIAL
EQUATION:
• In mathematics, the history of differential equations
traces the development of "differential equations" from
calculus, which itself was independently invented by
nglish physicist Isaac Newton and German mathematician
Gottfried Leibniz.
• The history of the subject of differential equations, in
concise form, from a synopsis of the recent article “The
History of Differential Equations, 1670-1950” “Differential
equations began with Leibniz, the Bernoulli brothers,
and others from the 1680s, not long after Newton’s
‘fluxional equations’ in the 1670s.”

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

5. Application of Ordinary
Differential Equation

6. Application of ODE
1. Newton’s Law of Cooling
2. Beam
3. Physical Application
4. Radio Active Elements
5. Electrical Circuits
6. MODELLING FREE MECHANICAL OSCILLATIONS
7. No Damping
8. Light Damping
9. Heavy Damping
10. MODELLING FORCED MECHANICAL OSCILLATIONS
11. COMPUTER EXERCISE OR ACTIVITY
12.MODELLING WITH FIRST-ORDER EQUATIONS

7. Physical Application of ODE

8. F= ma
THE RATE OF CHANGE IN MOMENTUM
ENCOUNTERED BY A MOVING OBJECT IS
EQUAL TO THE NET FORCE APPLIED TO IT.
IN MATHEMATICAL TERMS,
NEWTON’S SECOND LAW

9. NEWTON’S SECOND LAW

10. Newton’s law of cooling
Law: The rate of change of the temperature of an
object is proportional to the difference between its
own temperature and the temperature of its surroundings.
Therefore,
dθ / dt = E A (θ – θr ) ; E- A constant that depends upon
the object , A – surface area, θ – A certain
temperature, θr – Room/ ambient temperature or the
temperature of the surroundings.

11. RADIOACTIVE HALF-LIFE
dN/dt = kN
• A stochastic (random) process
• The RATE of decay is dependent upon the
number of molecules/atoms that are there
• Negative because the number is decreasing
• K is the constant of proportionality

12. Decay of Elements

13. Macaulay’s method
The differential equation of bending
becomes:
importantly, without expanding the term
into square brackets:
this expression can be integrated twice,

14. Beams