Benjamin's Maths World

Deriving the Green’s Function of a linear ODE using Laplace Transform and then employing that to solve the equation.

Utilizing Laplace Transform to solve ODEs with a source term that now contains step/delta functions.

Applying the method of Laplace Transform to convert ODEs into algebraic problems for easier solving.

The definition of Convolution \( [f*g](x) = \int_{-\infty}^{\infty} f(u)g(x-u) du \) and Convolution Theorem for Laplace Transform.

Heaviside Step Function and Dirac Delta Function, two shifting theorems, derivative of Laplace Transform.

The definition of Laplace Transform \( \mathscr{L}[f](s) = \int_0^{\infty} e^{-sx} f(x) dx \) and some notable Laplace Transform values.

Confluent Hypergeometric Equation by combining the two singularities \( x = 1,\infty \) of Hypergeometric Equation.

Hypergeometric equation at \(x = \infty\) and how any second-order linear ODEs with 3 regular singularities can be made into such a form.

Some other properties related to the Hypergeometric Function.

Hypergeometric Equation \( x(1-x) y^{\prime\prime} + [c-(a+b+1)x]y’ -aby = 0 \) and the corresponding Hypergeometric Function \( F(a,b,c;x) \).