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1. MA8353-TRANSFORMS
AND
PARTIAL DIFFERENTIAL EQAUTIONS

2. UNIT I

3. PARTIAL
DIFFERENTIAL
EQUATION

4. HISTORY OF DIFFERENTIAL
EQUATIONS
ISSAC NEWTON
GOTTFRIED LEIBNIZ
I
In Mathematics, history of
Differential equations the traces
the development of “Differential
Equations” from calculus, itself
independently invented by english
Physicist ISSAC NEWTON and
German MAthematician
GOTTFRIED LEIBNIZ

5. INDEPENDENT AND DEPENDENT VARIABLES
Real life example:
● Age of a person is a independent variable as we cannot increase or decrease
our age
● How tall you are at different age is a dependent variable
Dependent variable
Independent variable

6. TYPES OF DIFFERENTIAL EQUATIONS
DIFFERENTIAL
EQUATIONS
ORDINARY DIFFERENTIAL
EQUATIONS
PARTIAL DIFFERENTIAL
EQUATIONS

7. DIFFERENTIAL EQUATIONS
An equation with one or more terms that involves derivatives
of dependent variable with respect to an independent variable is
known as Differential Equations
Example:

8. ORDINARY DIFFERENTIAL EQUATIONS
An Ordinary Differential equation is Differential
Equation which depends on only one independent
variable
Example:

9. PARTIAL DIFFERENTIAL EQUATIONS
Partial Differential Equation is a Differential
Equation in which the dependent variables depends on
two or more independent variable
Example:

10. ORDER OF PARTIAL DIFFERENTIAL EQUATIONS
The Order of the PDE is the order of the highest order Partial
derivative occuring in the equation
DEGREE OF PARTIAL DIFFERENTIAL EQUATION
The degree of the highest derivative is the degree of the PDE
Example:
Here Order=3 and Degree=2 as the highest derivative is of order
3 and exponent raised to highest derivative is 2

11. Linear Partial Differential Equation
A PDE is said to be linear if the dependent variable and the partial
derivatives occur in the first degree and there is no product of partial
derivatives or product of derivative and dependent variable
Example:
1. is a second order linear equation
2. is a second order non-linear equation

12. HOMOGENEOUS & NON-HOMOGENEOUS LINEAR PDE
A Linear PDE in which all the partial derivatives are of same order is called a
homogeneous linear PDE otherwise it is non-homogeneous linear PDE
The general form is
--------------------(1)
Where are constants

14. SOLVING HOMOGENEOUS LINEAR PDE OF SECOND AND HIGHER
ORDER WITH CONSTANT COEFFICIENTS
Denote then equation (1) can be written as

15. Roots of Auxiliary equation Complementary Function