Unit3 A Differential Equations of First Order 2130002

Unit-3 A Differential Equations of First Order 2130002 – Advanced Engineering Mathematics Humanities & Science Department A. E. M. (2130002) Darshan Institute of engineering & Technology

Introduction Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 2

Definitions Ø Types of Differential Equations Ordinary Differential Equation Partial Differential Equation A. E. M. (2130002) Darshan Institute of engineering & Technology 3

Types of Differential Equations Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 4

Types of Differential Equations Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 5

Order of DE Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 6

Degree of DE Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 7

M-1 Examples on Order and Degree A. E. M. (2130002) Darshan Institute of engineering & Technology 8

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Types of solution Ø General solution A solution of a differential equation in which the number of arbitrary constants is equal to the order of the differential equation, is called the General solution or complete integral or complete primitive. Ø Particular solution The solution obtained from the general solution by giving a particular value to the arbitrary constants is called a particular solution. A. E. M. (2130002) Darshan Institute of engineering & Technology 11

Linear and Nonlinear DE Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 12

Types of First Order and First Degree DE ü Variable Separable Equation ü Homogeneous Differential Equation ü Linear(Leibnitz’s) Differential Equation ü Bernoulli’s Equation ü Exact Differential Equation A. E. M. (2130002) Darshan Institute of engineering & Technology 13

Variable Separable Method Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 14

M-2 Examples on Variable Separable DE A. E. M. (2130002) Darshan Institute of engineering & Technology 15

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Reducible to variable separable Eq. Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 22

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Leibnitz’s (linear) Equation Form - 1 Form -2 Form of DE Integrating factor Solution A. E. M. (2130002) Darshan Institute of engineering & Technology 26

M-3 Examples on Liebnitz’s DE A. E. M. (2130002) Darshan Institute of engineering & Technology 27

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Bernoulli’s Differential Equation Ø OR A. E. M. (2130002) Darshan Institute of engineering & Technology 36

Process to reduce the Bernoulli’s DE into Linear DE Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 37

Process to reduce the Bernoulli’s DE into Linear DE Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 38

Process to reduce the Bernoulli’s DE into Linear DE Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 39

Process to reduce the Bernoulli’s DE into Linear DE Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 40

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Exact Differential Equation Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 55

M-5 Examples on Exact DE A. E. M. (2130002) Darshan Institute of engineering & Technology 56

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Non Exact Differential Equation Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 64

Standard rules for finding I. F. Condition Type of equation I. F. Homogeneous Non Homogeneous - - A. E. M. (2130002) Darshan Institute of engineering & Technology 65

Non Exact Differential Equation Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 66

Homogeneous Differential Equation Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 67

Non-Homogeneous Differential Equation Ø A. E. M. (2130002) Darshan Institute of engineering & Technology 68

M-6 Examples on Non-Exact DE A. E. M. (2130002) Darshan Institute of engineering & Technology 69

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Orthogonal Trajectory Ø Trajectory A Curve which cuts every member of a given family of curves according to some definite rule is called trajectory. Ø Orthogonal Trajectory A curve which cuts every member of a given family at right angles is a called an Orthogonal Trajectory. A. E. M. (2130002) Darshan Institute of engineering & Technology 86

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M-7 Examples on Orthogonal Trajectory A. E. M. (2130002) Darshan Institute of engineering & Technology 88

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