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# module -4.ppt

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### module -4.ppt

1. 1. Calculus And Laplace transform (MAT 1001) (Online Course) By Ms. Sana Afreen SASL – Mathematics VIT , Bhopal University.
2. 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. 3. Introduction of Differential equation
4. 4. ORDER AND DEGREE OF A DIFFERENTIAL EQUATION
5. 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. 6. SOLUTION OF A DIFFERENTIAL EQUATION
7. 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. 8. 1. VARIABLES SEPARABLE
9. 9. HOMOGENEOUS DIFFERENTIAL EQUATIONS
10. 10. EQUATIONS REDUCIBLE TO HOMOGENEOUS FORM
11. 11. EXERCISE 3.1 Separation of variable & Homogeneous Equation
12. 12. LINEAR DIFFERENTIAL EQUATIONS
13. 13. EXERCISE 3.5 Linear Differential Equation
14. 14. EQUATIONS REDUCIBLE TO THE LINEAR FORM (BERNOULLI EQUATION)
15. 15. EXACT DIFFERENTIAL EQUATION
16. 16. Application of First order & Linear Differential Equation 1.ELECTRICAL CIRCUIT 2.Newton Law of Cooling 3.Population Decay
17. 17. LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH CONSTANT COEFFICIENTS
18. 18. COMPLETE SOLUTION= COMPLEMENTARY FUNCTION+ PARTICULAR INTEGRAL METHOD FOR FINDING THE COMPLEMENTARY FUNCTION
19. 19. EXERCISE 3.6 Second order Linear Differential Equation
20. 20. RULES TO FIND PARTICULAR INTEGRAL
21. 21. EXERCISE 3.7 Second order Linear Differential Equationwith cons. Coefficient
22. 22. METHOD OF VARIATION OF PARAMETERS
23. 23. EXERCISE 3.8 Second order Linear Differential Equation Variationof Parameter
24. 24. CAUCHY EULER HOMOGENEOUS LINEAR EQUATIONS
25. 25. EXERCISE 3.9 CAUCHY EULER HOMOGENEOUS LINEAR EQUATIONS