The document covers the history and classification of differential equations, beginning with Leibniz's work in 1675 and Newton's classification of first-order equations. It defines ordinary differential equations (ODEs) and partial differential equations (PDEs), providing examples and explaining concepts such as order and degree. The order is determined by the highest derivative present, while the degree refers to the power of the highest order derivative in a polynomial 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