Ode

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

7. SOLUTION OF A DIFFERENTIAL EQUATION

8. 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

9. 1. VARIABLES SEPARABLE

12. HOMOGENEOUS DIFFERENTIAL EQUATIONS

15. EQUATIONS REDUCIBLE TO HOMOGENEOUS FORM

20. EXERCISE 3.1 Separation of variable & Homogeneous Equation

22. LINEAR DIFFERENTIAL EQUATIONS

27. EXERCISE 3.5 Linear Differential Equation

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

34. EXACT DIFFERENTIAL EQUATION

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

44. LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH CONSTANT
COEFFICIENTS

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

48. EXERCISE 3.6 Second order Linear Differential Equation

49. RULES TO FIND PARTICULAR INTEGRAL

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

59. METHOD OF VARIATION OF PARAMETERS

63. EXERCISE 3.8 Second order Linear Differential Equation Variationof
Parameter

64. CAUCHY EULER HOMOGENEOUS LINEAR EQUATIONS

68. EXERCISE 3.9 CAUCHY EULER HOMOGENEOUS LINEAR EQUATIONS