Mathematics is a crazy and interesting subject.

Taylor and Maclaurin invented approximate polynomial series for continuous functions (and Laurrent in complex analysis), Fourier invented series of circular functions for both continuous and discontinuous functions, later the idea was extended to capture special functions (Bessel, Legendre, Laguerre, Chebyshev etc)

Sturm and Liouville woke up and told them all this things belong to just one concept: weight and orthogonality.

When Chebyshev, Bessel, Legendre, Hermit, Laguerre. Mathieu started throwing special functions with special ability and properties around, Frobenius came along and created one solution for all. and one equation called the father of all: Hypergeometric equation and all of the special functions became special cases of the hypergeometric function.

Then we have the transformers. You can't solve a problem in time space? take it to Laplace space (Heaveside) or frequency space (Fourier) or the Zeta space. And how many spaces do we have? countless. Its all a matter of Kernel.

With transforms, no matter the order of differentiation in the equation, all will be gone with just a single integration. Green came along and created a mega concept of operator methods in differential calculus. You can solve any differential equation by convolution of the right kernel with whatever boundary condition you have. To understand this, imagine you want to solve a linear system of equations: Ax = b for x. you performed an LU factorization of A and that's it, no matter the b you are given, you can always get x by using the same L and U over and over (x = (b/L)/U). You don't need to repeat the LU factorization again. In differential calculus, ones you find the green function which is a function of the domain once, you don't need to solve that again. you can be convoluting for different boundary conditions.

With Green, everything Laplace, Fourier, even Newton, Gauss, Leibniz. did in integration, all the fall under one concept: finding Kernel. Even the normal integration we learnt in school has a Kernel: It is hidden and equal to 1.