MAT201 / Mathematics

Syllabus

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M1: Ordinary differential equations

M2: Partial Differential Equations

Formation of Partial Differential Equations
First Order Partial Differential Equations [Video 1-5 in Playlist]
- Solutions of first-order linear PDEs
Second Order Partial Differential Equations [Video 6 onwards in Playlist]
- Homogeneous and non-homogeneous linear PDEs
- Complementary function and particular integral method (constant coefficients)
Non-linear Partial Differential Equations of First Order
- Charpit’s method
Separation of Variable Method
- Solution of wave and heat equations

M3: Laplace Transform and Fourier series

Laplace Transform
- Introduction and Laplace transform of elementary functions
- Properties
  - Change of scale property
  - Second shifting property
- Laplace transform of derivatives
- Inverse Laplace transform and its properties
- Convolution theorem
- Applications of Laplace Transform to Solve ODEs
Fourier Series
- Introduction to Fourier series
- Fourier series for discontinuous functions
- Fourier series for even and odd functions
- Half-range series

Function of a Complex Variable
Differentiation and Analytic Functions
- Cauchy-Riemann equations
- Harmonic functions and finding harmonic conjugates
Elementary Analytic Functions
- Exponential
- Trigonometric
- Logarithmic
Conformal Mappings
- Mobius transformations and their properties

Contour Integrals
Theorems (Without Proof)
- Cauchy-Goursat theorem
- Cauchy Integral formula
- Liouville’s theorem
- Maximum-Modulus theorem
Series Representations
- Taylor’s series
- Laurent’s series
Analysis of Analytic Functions
- Zeros
- Singularities
Residues and Applications
- Cauchy Residue theorem (without proof)
- Evaluation of definite integrals involving sine and cosine
- Evaluation of certain improper integrals