WEEK 1
First order ODE: Introduction to Differential Equations (ODEs); terminology and classification; formation and solution of ODEs; initial and boundary Value problems. Method for first order ODE: Separable ODEs; Homogeneous ODEs;
WEEK 2
Linear ODEs; Exact ODEs; Bernoulli Equations. (Existence & Uniqueness of Solutions for Initial Value Problem). Applications of first order ODEs.
WEEK 3
Linear Second Order ODEs: Solution of non-homogenous equations. Method of undetermined coefficients. Method of the undetermined coefficients to higher order ODE’s up to fourth order.
WEEK 4
Solution of homogeneous Linear ODEs with Constant Coefficients methods of undetermined coefficients; methods of variation of parameters.
WEEK 5
Applications of second order ODE. Laplace Transforms: Definition of the Laplace Transforms, derivation of Laplace transforms for standard elementary functions. Linearity property
WEEK 6
Solution of homogeneous Linear ODEs with Constant Coefficients methods of undetermined coefficients; methods of variation of parameters.
WEEK 7
First shifting theorem, multiplication by t^n, Laplace transforms of unit step functions, Laplace transforms of Delta Dirac functions and periodic functions;
WEEK 8
System of differential equation using Laplace transform.
WEEK 9
Fourier Series: Periodic function. Even and odd functions. Basic properties
WEEK 10
Fourier series for periodic functions. Fourier series for even function. Fourier series for odd functions.
WEEK 11
Half range series, Cosine Fourier Series, Sine Fourier Series; Complex form of the Fourier integral.
WEEK 12
Partial Differential Equations (PDEs): Basic Definition and Terminologies. Method of Separation of variables. Solution of the Heat Equation. Rod with insulated end points. Rod with zero temperature at endpoints.
WEEK 13
Method of Separation of variables. Solution of the Wave Equation. String with fixed endpoints. String with free endpoints. Method of d’Alembert for solving wave equations.
WEEK 14
Solution of the Laplace Equation. Prescribed temperature at the boundaries – type A. Prescribed temperature at the boundaries – type B.