Series solutions for second order linear differential equations.
Review on power series.
Series solutions near ordinary points.
Series solutions near regular singular points (Case 1: When the indicial roots do not differ by an integer.) (Case 2: When the indicial roots are equal.) (Case 3: When the indicial roots differ by an integer.).
Series solutions near infinity.

Legendre polynomials
Solution of the Legendre equation
Rodrigue formula
Recurrence formula for Legendre polynomials.
Generating function for Legendre polynomials
Orthogonality of Legendre polynomials
Series of Legendre polynomials

Fourier series
Expansion of periodic functions in Fourier series.
Half ranged Fourier sine and cosine series.

Classical equations and BVPs solved by method of separable variables
Review on separable variables method
One-dimensional heat equation
One-dimensional wave equation
Two-dimensional Laplace equation

Q1 : What do students learn in series solutions for second order linear differential equations?
A1 : Students will learn the power series method to solve the second order linear differential equations using an ordinary point. Students will learn the Frobenius method to solve the second order linear differential equations using a regular singular point.

Q2 : What do students learn in Legendre polynomials?
A2 : Students will learn to solve the first series solution of the Legendre equations. Students will learn to find the Legendre polynomials by using various methods.

Q3 : What do students learn in Fourier series?
A3 : Students will learn to find the Fourier series expansion of periodic functions.

Q4 : What do students learn in classical equations and BVPs solved by method of separable variables?
A4 : Students will learn to solve heat equation, wave equation and Laplace equation by using the method of separable variables and Fourier series.