History of Differential Equation Ordinary Differential Equation With Examples Partial Differential Equations With Examples Order & Degree of Differential Equation

1. Differential
Equations
History of Differential Equations
Definition & Types of Differential Equation
Order & Degree of Differential Equations

2. History of Differential Equations:
According to some Historians of Mathematics, the
study of differential equations began in 1675, when
Gottfried Leibniz (1646-1716) wrote the equation:
𝑥𝑑𝑥 =
𝑥2
2

3. History of Differential Equations:
The search for general methods of integrating
differential equations began when Isaac Newton
(1642-1727) classified first order differential
equations into three classes:
I.
𝑑𝑦
𝑑𝑥
= 𝑓 𝑥
II.
𝑑𝑦
𝑑𝑥
= 𝑓(𝑥, 𝑦)
III. 𝑥
𝜕𝑢
𝜕𝑥
+ 𝑦
𝜕𝑢
𝜕𝑦
= 𝑢

4. Differential Equations & their
Classification:
Definition:
An equation involving derivatives or differential of one or
more dependent variable with respect to one or more
independent variables is called a differential equation.
Examples:
I.
𝑑𝑦
𝑑𝑥
+ 𝑥2 𝑦 = 𝑒 𝑥−2
II. 2
𝑑2 𝑦
𝑑𝑥2 +
𝑑𝑦
𝑑𝑥
− 𝑦 = 𝑐𝑜𝑠𝑥
III.
𝑑3 𝑦
𝑑𝑥3 + 𝑥𝑦
𝑑𝑦
𝑑𝑥
2
= 0

5. Differential Equations &
their Classification:
Types of Differential Equation:
 Ordinary differential equation
 Partial differential equation

6. Ordinary Differential
Equation:
Definition:
An ordinary differential equation is one in which there
is only one independent variable.
Hence a differential equation involving ordinary
derivatives of one or more dependent variables with
respect to a single independent variable is called an
ordinary differential equation

7. Ordinary Differential
Equation:
Examples:
I. 2
𝑑𝑥
𝑑𝑡
−
𝑑𝑦
𝑑𝑡
− 3𝑦 = 2
II.
𝑑3 𝑦
𝑑𝑡3 + 𝑥𝑦
𝑑𝑦
𝑑𝑡
2
= 0
III.
𝑑2 𝑦
𝑑𝑥2 + 9
𝑑𝑦
𝑑𝑥
− 𝑦 = 𝑠𝑖𝑛𝑥

8. Partial Differential
Equation:
Definition:
A differential equation involving partial derivatives of
one or more dependent variable with respect to one or
more independent variables is called a partial
differential equation.

9. Partial Differential
Equation:
Example:
I.
𝜕𝑢
𝜕𝑥
+
𝜕𝑢
𝜕𝑦
= 0
II.
𝜕2 𝑢
𝜕𝑥2 +
𝜕2 𝑣
𝜕𝑦2 = 0
III.
𝜕𝑦
𝜕𝑥
+
𝜕𝑢
𝜕𝑥
= 0

10. Order of Differential
Equation:
Definition:
The order of a differential equation is the order of the
highest derivative appearing in the equation.
Examples:
I.
𝑑3 𝑦
𝑑𝑡3 + 𝑥𝑦
𝑑𝑦
𝑑𝑡
2
= 0
is of order 3 and it is ordinary differential equation.
I.
𝜕𝑢
𝜕𝑥
2
+
𝜕𝑢
𝜕𝑦
= 𝑐𝑜𝑠𝑥
is of order 1 and it is partial differential equation.

11. Degree of Differential
Equation:
Definition:
The degree of differential equation is represented by
the power of the highest order derivative in the given
differential equation.
The differential equation must be a polynomial equation
in derivatives for the degree to be defined.
Examples:
I.
𝑑4 𝑦
𝑑𝑥4 +
𝑑2 𝑦
𝑑𝑥2
2
− 3
𝑑𝑦
𝑑𝑥
+ 𝑦 = 9 is of degree 2
II.
𝑑2 𝑦
𝑑𝑥2 + 𝑐𝑜𝑠
𝑑𝑦
𝑑𝑥
= 9𝑥𝑦 is of degree not defined