Differential Equations and its real meaning through real life problem.

1. V.V. VANNIAPERUMAL COLLEGE FOR WOMEN
Introduction of Differential Equation
Dr. T. Anitha
Assistant Professor
Department of Mathematics(SF)

2. A B
Distance traveled from A to B is S=100m
Time taken to travel t=10s

3. A B
Velocity=
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑖𝑚𝑒
=
100
10
= 10𝑚/𝑠

4. A B
Velocity =
𝑆 𝐷 −𝑆(𝐶)
𝑡𝐷−𝑡𝐶
C D

5. time
distance
a x
Y=f(t)
f(a)
f(x)
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
𝑓(𝑥) − 𝑓(𝑎)
𝑥 − 𝑎

6. time
distance
a x
Y=f(t)
f(a)
f(x)
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = lim
𝑥→𝑎
𝑓(𝑥) − 𝑓(𝑎)
𝑥 − 𝑎
=
𝑑𝑦
𝑑𝑥

7. Differentiation
Differentiation is how rapidly that quantity changes with respect to change
in another quantity.
Differential Equation
Differential equation is an equation in which differential coefficient occur.

8. Examples:
Differential equations are of two types:
 Ordinary differential equation
Partial differential equation
1. y’ + 2𝑦 + 3𝑥 = 0
2.
𝑑2𝑦
𝑑𝑥2 + 𝑚2
𝑦 = 0

9. Dependent and independent variable
S
𝐴𝑟𝑒𝑎 = 𝑆2
S+d
𝐴𝑟𝑒𝑎 = 𝑆 + 𝑑 2
So Area is a function of side (s)
Therefore area 𝐴 = 𝑓(𝑆) = 𝑆2
The perimeter P=f(S)=4s
Hence in the variables A, P, S,
A and P are dependent variables and
S is the independent variable.

10. Dependent and independent variable
L
Area=BL
L+d
Area=(L+d)(B+r)
B
B+r
So Area of a rectangle
is a function of length (L) and breath (B)
Therefore area A=f(B,L)=BL
The perimeter P=f(B,L)=2(B+L)
Hence, in the variables A, P, B, L
A and P are dependent variables and
L, B are the independent variables.

11. Ordinary differentiation
 y=f(x)

𝑑𝑦
𝑑𝑥
(rate of change of y w.r.t x)
Partial differentiation
 y=f(x,z)

𝜕𝑦
𝜕𝑥
(rate of change of y w.r.t x)

𝜕𝑦
𝜕𝑧
(rate of change of y w.r.t z)

12. Ordinary differential equation
An ordinary differential equation is an equation in which a single independent
variable enters either explicitly or implicitly
Example
1.
𝑑𝑦
𝑑𝑥
= 2 𝑠𝑖𝑛𝑥 (Here x is independent variable and y is dependent variable)
2.
𝑑2𝑦
𝑑𝑡2 = 𝑚2𝑥 (Here t is independent variable and x, y are dependent variable)

13. Partial differential equation
A partial differential equation is an equation in which at least two or more independent
variables enter and partial coefficient occurring in them.
Example
x
𝜕𝑧
𝜕𝑥
+ 𝑦
𝜕𝑧
𝜕𝑦
= 𝑧 (Here x,y are independent variable and z is dependent variable)

14. Thank you