TITLE – ORDINARY DIFFERENTIAL EQUATIONS
AND THEIR APPLICATIONS
Report / Dissertation of Minor Research Project STUDY OF ORDINARY
DIFFERENTIAL EQUATIONS IN MATHEMATICS
in Subject MATHEMATICS
Submitted for the Award of the Degree of B.Sc. (Bachelor of science)
Aligned with NEP 2020 By 2024-25
STUDENT’S NAME KARAN NISHAD
Univ. Roll No. 22040100150167
Under the Supervision of
MR. BADRI VISHAL SINGH
SUPERVISOR’S NAME
MR. BADRI VISHAL SINGH
K.S.SAKET PG COLLEGE, AYODHYA Dr. R.M.L. AVADH
UNIVERSITY, AYODHYA DECLERATION CERTIFICATE
I KARAN NISHAD S/o MUNNA NISHAD & GUDIYA NISHAD certify that
the work embodied in this minor research project is my own bonafied work carried out by me
under the supervision of ………………………….. The work embodied in this report /
dissertation of minor research work has not be submitted earlier elsewhere except where due
acknowledgment has been made
in the text.
I, hereby declare that I have faithfully acknowledged, given credit to and refereed to the
workers wherever their works have been cited in the text and the body of report. I further certify
that I have not willfully lifted up some other’s work. para, text, data, results, etc. reported in the
journals, books, magazines, reports, dissertations, thesis, etc or available at websites and
included them in this report / dissertation and cited as my own work.
Date ………………… < Sign of Scholar >
Place ………………... KARAN NISHAD
Univ. Roll No. 22040100150167
UNDERTAKING FORM THE GRADUATE SCHOLAR
I hereby declare that, I …………………………………………………………. have
completed the minor research project work as per NEP 2020 on the title STUDY OF VECTOR
SAPCES AND SUBSPACES IN MATHEMATICS
under the supervision of ………………………………. for the
degree B.Sc. of Dr. R.M.L. Avadh University, Ayodhya.
This is my own work and I have not submitted it earlier elsewhere.
< Sign of Scholar>
KARAN NISHAD Date …………………
Place ………………... Univ. Roll No. 22040100150167
NAME OF SUPERVISOR NAME OF DEPARTMENT K.S.SAKET
PG COLLEGE AYODHYA
CERTIFICATE
This is to certify that research work embodied in this report / dissertation of minor research
project entitle” STUDY OF ORDINARY DIFFERENTIAL EQUATIONS AND THEIR
APPLICATION IN MATHEMATICS ” submitted for the award of the degree B.Sc. as per
NEP 2020 has been carried out by KARAN NISHAD under my supervision.
To the best of my knowledge and belief, this work is original and has not been
submitted so far in part or in full for the award of any degree or diploma elsewhere.
Date …………………. <Sign of Supervisor>
Place………………………..
Ordinary Differential Equations and Their Applications
1. Introduction to Differential Equations
- Definition of a differential equation
- Order and degree of a differential equation
- Types of differential equations:
- Ordinary vs Partial
- Linear vs Non-linear
- Homogeneous vs Non-homogeneous
2. First-Order Differential Equations
- Separable Variables
- Homogeneous Equations
- Linear First-Order Equations (Integrating Factor method)
- Exact Differential Equations
- Applications:
- Population growth & decay
- Newton's law of cooling
- Mixing problems
- Radioactive decay
3. Second and Higher-Order Linear ODEs
- General form: a_n(x)y + ... + a(x)y = g(x)
- Homogeneous linear equations with constant coefficients
- Method of undetermined coefficients
- Variation of parameters
Ordinary Differential Equations and Their Applications
4. Systems of First-Order Linear ODEs
- Matrix method
- Eigenvalues and eigenvectors
- Solution using diagonalization
5. Laplace Transform Method
- Definition and properties
- Solving initial value problems
- Inverse Laplace Transforms
- Applications:
- Electric circuits
- Mechanical systems
- Control systems
6. Series Solutions of ODEs
- Power series method
- Frobenius method
- Regular and singular points
7. Numerical Methods for ODEs
- Euler's method
- Runge-Kutta methods
- Applications when analytical solutions are difficult
Ordinary Differential Equations and Their Applications
8. Applications of ODEs
- Physics: Motion, Oscillations, Circuits
- Biology: Population dynamics, Spread of diseases
- Chemistry: Reaction rates
- Engineering: Heat transfer, Vibration analysis
- Economics: Growth models, Interest rates
