## Topic outline

• ### General

 Lecturer : WAN RUKAIDA WAN ABDULLAHYUDARIAH MOHAMMAD YUSOFNUR ARINA BAZILAH AZIZZUHAILA ISMAILSHAZIRAWATI MOHD PUZI Semester : I, 2016/2017

Synopsis :

This is an introductory course on differential equations. Topics include first order ordinary differential equations (ODEs), linear second order ODEs with constant coefficients up to fourth order, the Laplace transform and its inverse, Fourier series, and partial differential equations (PDEs). Students will learn how to classify and solve first order ODEs, use the techniques of undetermined coefficients, variation of parameters and the Laplace transform to solve ODEs with specified initial and boundary conditions, and use the technique of separation of variables to solve linear second order PDEs and the method of d’Alembert to solve wave equation.

• ### Topic 1

First order ordinary differential equations

• Definition and classification of differential equations. Basic ideas; solutions of differential equations, initial and boundary value problems.
• Solving 5 types of equations:

1. Separable equation
2. Linear equations
3. Homogeneous equations
4. Exact equations
5. Bernoulli equations

• Applications such as law of cooling , the free fall and chemical reactions .

• ### Topic 2

Linear second order ordinary differential equations with constant coefficients

• Second order homogeneous differential equations.
• Solution of non- homogenous equations.
1. Method of undetermined coefficients.
2. Method of the undetermined coefficients to higher order ODE’s up to fourth order,
3. Method of variation of parameters.
• Applications of second order differential equations: mechanical vibrations, damped and undamped free vibrations, and electrical circuits, circuits with and without impedance/resistance

• ### Topic 3

Laplace transforms

Definition of Laplace transforms, derivation of Laplace transforms for standard elementary functions. Linearity property, first shifting theorem, multiplication by Laplace transforms of unit step functions, Laplace transforms of Delta Dirac functions and periodic functions; Second shifting Theorem, Laplace transforms of the derivatives. Inverse Laplace transforms and Convolution theorem; Solving initial value problems ( IVP), boundary value problems (BVP) and system of differential equation using Laplace transform.

• ### Topic 4

Fourier series:

Even and odd functions. Fourier series for periodic functions. Fourier series for even and odd functions, Half-range Fourier series,

• ### Topic 5

Partial differential equations.

Basic concepts, classifications. Method of d’Alembert for solving wave equations. Method of separation of variables for solving heat equation (consolidation theory), wave equations. Method of separation of variables for solving Laplace equations.