Differential Equations

Course Title: Differential Equations

Course No: BDS205

Nature of the Course: THEORY

Semester: 3

Full Marks: 45 + 30

Pass Marks: 18 + 12

Credit Hours: 3

Course Description

This course introduces differential equations and their applications. It covers first, second and higher order differential equations. It also includes system of first order linear equations, series solution of second order linear equations, Laplace transform and their applications.

Course Objectives

After the successful ending of this course the student will be able to Solve first, second and higher order differential equation. Solve system of first order linear equations. Apply Laplace transform in solving initial value problems.

Course Contents

1. First Order Differential Equations

12 hrs

1.1. Differential equations and its classification

1.2. Solution of differential equations

1.3. Some basic mathematical models and direction fields

1.4. Separable equations

1.5. Homogeneous differential equations

1.6. Exact equations

1.7. Linear equations

1.8. Bernoulli equations

1.9. Modeling with first order equations

1.10. Autonomous differential equations and population dynamics

1.11. Numerical approximations: Euler's method

1.12. Existence and uniqueness theorem

1.13. First order higher degree differential equations

2. Second and Higher Order Linear Equations

12 hrs

2.1. Second order linear equations

Homogeneous equations with constant coefficients

Solution of linear homogeneous equations; Wronskian

Real and complex roots of the characteristic equation

Reduction of order

Non-homogeneous equations; Methods of undetermined coefficients and variation of parameters

2.2. Higher order linear equations

General theory of nth order linear equations

Homogeneous differential equations with constant coefficients

Methods of undetermined coefficients and variation of parameters

3. Series Solution of Second Order Linear Equations

8 hrs

3.1. Series solutions near an ordinary point

3.2. Regular and irregular singular points

3.3. Euler equations

3.4. Series solutions near a regular singular point

3.5. Bessel's equation

4. System of First Order Linear Equations

6 hrs

4.1. Systems of linear algebraic equations; Linear independence, eigenvalues, eigenvectors

4.2. Basic theory of first order linear equations

4.3. Homogeneous linear system with constant coefficients

4.4. Complex eigenvalues

5. Laplace Transform

10 hrs

5.1. Laplace transform and its properties

5.2. Inverse Laplace transform and its properties

5.3. Step functions

5.4. Differential equations with discontinuous forcing functions

5.5. Impulse functions

5.6. Convolution integral

5.7. Application of Laplace transform to initial value problems

Reference Books

1.Boyce, W.E., DiPrima, R.C and Meade D.B., Elementary Differential Equations and Boundary Value Problems, 9th edition., Wiley India Pvt. Ltd.
2.James C. Robinson, An Introduction to Ordinary Differential Equations, Cambridge University Press.
3.Shepley L. Rose (2010), Differential Equations, 3rd edition, James Wieley India.

Notes:

This syllabus follows the official Bachelor in Data Science curriculum of Tribhuvan University. In case of any doubt or revision, the university's published syllabus shall be considered authoritative.

Differential Equations

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