## PHY 223: Ordinary and Partial Differential Equations

3 credits | Prerequisites: PHY 220, 221

### Course rationale

This course is mandatory for physics major students. The appearance of differential equations can be found almost everywhere in physics. The physical laws are often expressed in terms of differential equations which we need to solve in different cases of interest. Therefore, it is very important to know the techniques to solve differential equations. This course will lay the foundation of the theory and techniques to solve differential equations: both ordinary and partial.

### Course content

Classification of Differential Equations: solutions of ODE, supplementary conditions, direction fields, existence and uniqueness theorems; First Order Differential Equations: linear equations, exact equations, integrating factors, substitutions, and transformations; applications of first-order differential equations, orthogonal and oblique trajectories; Higher Order Linear differential equations: linear differential operators, the basic theory of linear differential equations, homogeneous systems; Methods to Solve Second Order ODEs: reduction of order, nonhomogeneous equation, method of undetermined coefficients, variation of parameters, Euler-Cauchy differential equations, Green’s function; First Order PDEs: complete integral, general solution, Cauchy problems, method of characteristics for linear and quasi-linear equations, Charpit’s method for finding complete integrals, methods for finding general solutions; Second Order PDEs: classifications, reduction to canonical forms, characteristic curves; Boundary Value Problems Related to Linear Equations: applications of Fourier methods, mixed boundary conditions; Problems Involving Cylindrical and Spherical Symmetry: boundary value problems involving special functions, Transform Methods for Boundary Value Problems: Fourier transforms, nonhomogeneous equations, Green’s function.

### Course objectives

- Analyze and understand the theory of differential equations.
- Use different methods to solve ordinary differential equations.
- Apply the techniques to solve different physical problems.
- Use different techniques to solve partial differential equations.
- Apply integral transform and Green’s function to solve partial differential equations.

### References

- Differential Equations with Boundary Value Problems (8th edition) by Dennis G. Zill, and Warren S.Wright
- Elementary Differential Equations and Boundary Value Problem (12th edition) by William E. Boyce, Richard C. DiPrima, and Douglas B. Meade
- Partial Differential Equations: Analytical Solution Techniques (2nd edition) by Jirair Kevorkian
- Partial Differential Equations (2nd edition) by Lawrence C. Evans
- Introduction to Partial Differential Equations, by Peter Olver
- Partial Differential Equations of Mathematical Physics by S. L. Sobolev
- Mathematical Methods for Physicists (7th edition) by George B. Arfken, Hans J. Weber, and Frank E. Harris