1) Partial differential equations (PDEs) contain partial derivatives of an unknown function of two or more independent variables. 2) The order of a PDE is defined as the order of the highest partial derivative. The degree is the highest order partial derivative after rationalization. 3) Notations used include subscripts for partial derivatives with respect to independent variables like ∂z/∂x. 4) First order PDEs are classified as linear if linear in derivatives, semi-linear if linear in derivatives and coefficients depend only on variables, and quasi-linear or nonlinear if not linear in derivatives.

1. MADHAV INSTITUTE OF TECHNOLOGY & SCIENCE
Calculus & Optimization
Techniques (2240421)
Submitted to:
Dr. Dilip Kumar Mishra
(4th Semester)
Submitted by :
Shivam Mittal
0901MC221066
AIR- II year

2. Partial Differential
equation

3. Differential equation
• The equation which express the relationship between independent variables & dependent variables
and the derivatives of dependent variables w.r.t independent variables is called Differential eqn.
• Examples:
ⅆ𝑦
ⅆ𝑥
+ 𝑦 = 𝑥 ; here x is independent variable and y is dependent variable.
• 𝑬𝒙𝒂𝒎𝒑𝒍𝒆𝒔:
𝑑𝑦
𝑑𝑥
− 𝑦 = 𝑒𝑥
𝑑2
𝑦
𝑑𝑥2
− 5
𝑑𝑦
𝑑𝑥
+ 2𝑦 = cos( 𝑥)

4. Mainly there are two types of differential equation exist:
• Ordinary differential eqn. (ODE)
• Partial differential eqn. (PDE)
❖ Ordinary differential eqn. (ODE) :
• ODE involve one or more ordinary derivatives of unknown functions with respect to only one
independent variable.
)
cos(
2
5
:
2
2
x
y
dx
dy
dx
y
d
e
y
dx
dy
Examples
x
=
+
−
=
− y(x): unknown function
x: independent variable

5. Partial Differential equation
• An equation containing one or more partial derivatives of an unknown function of two or more
independent variables is known as a Partial Differential Equation.
• Examples:

6. Order & Degree of PDE
❖ Order of PDE:
• The order of the PDE is defined as the order of the highest partial derivative occurring in it.
❖ Degree of PDE:
• The degree of a PDE is defined as the degree of the highest order partial derivative which occurring in
it after the eqn. has been rationalized i.e. made free from radicals or fractions.

7. • Examples:

8. Notations used in PDE
(i) When we consider two independent variables 𝑥 and 𝑦 and one dependent variable 𝑧. Then, we use
the following notations in the PDEs.
(ii) When we consider 𝑛 independent variables 𝑥1, 𝑥2, 𝑥3, … … , 𝑥𝑛 and one dependent variable 𝑧.
Then, we use the following notations in the PDEs.
(iii) Partial differentiations are also denoted as :

9. Classification of First order PDE
(i)Linear PDE :
A first order equation 𝑓(𝑥, 𝑦, 𝑧, 𝑝, 𝑞) = 0 is said to be Linear PDE if it is linear in 𝑝, 𝑞 𝑎𝑛𝑑 𝑧,
that is, if the given equation is of the form 𝑃(𝑥, 𝑦)𝑝 + 𝑄(𝑥, 𝑦)𝑞 = 𝑅(𝑥, 𝑦)𝑧 + 𝑆(𝑥, 𝑦).
Examples:

10. Classification of First order PDE
(ii) Semi-linear PDE :
A first order equation 𝑓(𝑥, 𝑦, 𝑧, 𝑝, 𝑞) = 0 is said to be Semi-linear PDE if it is linear in 𝑝 𝑎𝑛𝑑 𝑞,
the coefficients 𝑝 𝑎𝑛𝑑 𝑞 are functions of 𝑥 𝑎𝑛𝑑 𝑦 only, that is, if the given equation is of the
form 𝑃(𝑥, 𝑦)𝑝 + 𝑄(𝑥, 𝑦)𝑞 = 𝑅(𝑥, 𝑦, 𝑧).
Examples:

11. Classification of First order PDE
(iii) Quasi-linear PDE :
A first order equation 𝑓(𝑥, 𝑦, 𝑧, 𝑝, 𝑞) = 0 is said to be Quasi-linear PDE if it is linear in 𝑝 𝑎𝑛𝑑 𝑞,
that is, if the given equation is of the form 𝑃(𝑥, 𝑦, 𝑧)𝑝 + 𝑄(𝑥, 𝑦, 𝑧)𝑞 = 𝑅(𝑥, 𝑦, 𝑧).
Examples:

12. Classification of First order PDE
(iv) Non-linear PDE :
A first order equation 𝑓(𝑥, 𝑦, 𝑧, 𝑝, 𝑞) = 0 is said to be Nonlinear PDE if it is not linear in
𝑝 𝑎𝑛𝑑 𝑞.
Examples:

13. Thank You