1.
Calculus And Laplace
transform (MAT 1001)
(Online Course)
By
Ms. Sana Afreen
SASL – Mathematics
VIT , Bhopal University.

2.
Module : 4 Differential equations
• Content:
• First order linear differential equations
• Second order linear differential equations with constant coefficients –
complementary function
• Method of undetermined coefficients
• Method of variation of parameters
• general solutions of homogeneous and non-homogeneous equations –
Cauchy-Legendre and Cauchy-Euler equations

3.
Introduction of Differential equation

4.
ORDER AND DEGREE OF A DIFFERENTIAL EQUATION

5.
FORMATION OF DIFFERENTIAL EQUATIONS
The differential equations can be formed by differentiating the ordinary equation and eliminating the arbitrary
constants.
Example 1. Form the differential equation by eliminating arbitrary constants, in the following cases and also write
downthe order of the differentialequationsobtained

6.
SOLUTION OF A DIFFERENTIAL EQUATION

7.
We will discuss the standard methods of solving the differential
equations of the following types:
(i)Equations solvable by separation of the variables.
(ii)Homogeneous equations.
(iii)Linear equations of the first order.
(iv)Exact differential equations.
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND FIRST DEGREE

8.
1. VARIABLES SEPARABLE

9.
HOMOGENEOUS DIFFERENTIAL EQUATIONS

10.
EQUATIONS REDUCIBLE TO HOMOGENEOUS FORM

11.
EXERCISE 3.1 Separation of variable & Homogeneous Equation

12.
LINEAR DIFFERENTIAL EQUATIONS

13.
EXERCISE 3.5 Linear Differential Equation

14.
EQUATIONS REDUCIBLE TO THE LINEAR FORM (BERNOULLI EQUATION)

15.
EXACT DIFFERENTIAL EQUATION

16.
Application of First order & Linear Differential Equation
1.ELECTRICAL CIRCUIT
2.Newton Law of Cooling
3.Population Decay

17.
LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH CONSTANT
COEFFICIENTS

18.
COMPLETE SOLUTION= COMPLEMENTARY FUNCTION+ PARTICULAR INTEGRAL
METHOD FOR FINDING THE COMPLEMENTARY FUNCTION

19.
EXERCISE 3.6 Second order Linear Differential Equation

20.
RULES TO FIND PARTICULAR INTEGRAL

21.
EXERCISE 3.7 Second order Linear Differential Equationwith cons.
Coefficient

22.
METHOD OF VARIATION OF PARAMETERS

23.
EXERCISE 3.8 Second order Linear Differential Equation Variationof
Parameter

24.
CAUCHY EULER HOMOGENEOUS LINEAR EQUATIONS

25.
EXERCISE 3.9 CAUCHY EULER HOMOGENEOUS LINEAR EQUATIONS