To provide a solid understanding of differential equations, it uses and wide applications. 

The main objective of this course is to provide the students with a range of ordinary differential equations that are essential to the solution of advanced problems encountered in the fields of applied mathematics, physics and engineering. In addition, this course is intended to prepare the students with mathematical tools and techniques that are required in advanced courses offered in the applied physics and engineering programs.To introduce students to the formulation, classification of differential equations and existence and uniqueness of solutions. To provide skill in solving initial value and boundary value problems. To develop understanding and skill in solving first and second order linear homogeneous and no homogeneous differential equations and solving differential equations using power series methods.

INTENDED LEARNING OUTCOMES

After completing this course, student should be able to understand the types of transformations and their practical use. Also, they would be able to know their importance in higher education in particular research. Furthermore the sutudents will be able to

1. Explain the concept of differential equation.
2. Classified the differential equations with respect to their order and linearity.
3. Explain the meaning of solution of a differential equation.
4. Express the existence-uniqueness theorem of differential equations.
5. Solve first-order ordinary differential equations.
6. Solve exact differential equations.
7. Solve Bernoulli , Ricaati and Clairautdifferential equations.
8. Model radioactive decay, compound interest, and mixing problems using first
   order equations.
9. Model population dynamics using first order autonomous equations.
10. Find solution of higher-order linear differential equations.
11. Express the basic existence theorem for higher- order linear differential equations.
12. Solve the homogeneous and non- homogeneous linear differential equations with constant coefficients.
13. Use the method "variations of parameters" to find to solution of higher-order linear differential equations with variable coefficients.
14. Solve the Cauchy-Euler equations.
15. Determine if a set of functions is linearly dependent or independent by
    definition and by using the Wronskian.
16. Construct a second solution of a differential equation from a known
    solution.
17. Find eigenvalues and eigen functions.
18. Use the eigenvalue-eigenvector method to find the general solution of first order
    linear 2 × 2 homogeneous systems with constant coefficients.
19. Solve simple eigenvalue problems of Sturm-Liouville type
20. Use power series methods to solve differential equations about ordinary
    points.
21. Use the Method of Frobenius to solve differential equations about regular singular points

COURSE CONTENTS

1. Preliminaries: Introduction and formulation, classification of differential equations
2. Existence and uniqueness of solutions
3. Introduction of initial value and boundary value problems
4. First order ordinary differential equations: Basic concepts, formation and solution of differential equations
5. Separable variables, Exact Equations, Homogeneous Equations, Linear equations, integrating factors
6.  Some nonlinear first order equations with known solution, differential equations of Bernoulli and Ricaati type,Clairaut equation
7. Modeling with first-order ODEs, Basic theory of systems of first order linear equations
8.  Homogeneous linear system with constant coefficients, Non homogeneous linear system,Second and higher order linear differential equations
9. Initial value and boundary value problems, Homogeneous and non-homogeneous equations,
10. Superposition principle, homogeneous equations with constant coefficients, Linear independence and Wronskian
11.  Nonhomogeneous equations, undetermined coefficients method, variation of parameters, Cauchy-Euler equation
12.  Modeling. Sturm-Liouville problems: Introduction to eigen value problem, adjoint and self adjoint operators
13.  Self adjoint differential equations, eigen values and eigen functions, Sturm-Liouville (S-L) boundary value problems, regular and singular S-L problems, properties ofregular S-L problems
14. Series Solutions: Power series, ordinary and singular points, types of singular points ,Existence of power series solutions
15. Frobenius theorem, Existence of Frobenius series solutions, solutions about singular points
16. The Bessel, Modified Bessel, Legendre and Hermite equations and their solutions.

Recommended Books

1. Dennis G. Zill and Michael R., Differential equations with boundary-value problems, 8th Edition (Brooks/Cole, 1997).
2. William E. Boyce and Richard C. Diprima, Elementary differential equations and boundary value problems, Seventh Edition John Wiley & Sons, Inc

Suggested Books

1. Arnold V. I., Ordinary Differential Equations, Springer, 1991.
2. Apostol T., Multi Variable Calculus and Linear Algebra, 2nd ed.,( John Wiley and sons, 1997).

RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS

Assignments would be given related topics. These may cover to interesting fields related to mathematics. Main purpose would be to learn about various kind of transformations and their importance in field.

ASSESSMENT CRITERIA

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam:   30

Final-Term Exam: 50

Key Dates and Time of class meeting

Monday : 8:00 am -9:30 am.    

Tuesday: 8:00am-9:30am.

Commencement of Classes: October 12, 2020                                                  

Mid Term Examination:            December 14-18, 2020                                                           

Final Term Examination           February 08-12, 2021                                              

Declaration of Result               February 19, 2021