Uploaded September 2017 | Updated September 2026, 2 weeks ago
In this video, I describe how to use Green's functions (i.e. responses to single impulse inputs to an ODE) to solve a non-homogeneous (Sturm-Liouville) ODE subject to ANY arbitrary input f(x). All this involves is integrating the Green's function with the input f(x) over the domain. The input, by the way, is the function that makes the ODE non-homogeneous.
This approach is made possible because of the orthogonality relation of the Sturm-Liouville Theorem and the fact that the eigenfunctions of a REGULAR Sturm-Liouville problem form a complete set.
By the way, at 2:05, I say that we can make the eigenvalue 'any real number we want'. I should clarify that and say that we can make the 'n' in lambda_n 'whatever integer we want'. Lambda is still restricted by the boundary conditions. Nonetheless, the point that a smaller set of functions (eigenfunctions of a REGULAR Sturm-Liouville Problem) can be used to describe a larger set of functions (all nice/smooth functions) still holds.
Questions/requests? Let me know in the comments! The oft-requested PDE version of my Green's Function video is coming out soon, so stay tuned!
Prereqs: Know the Sturm-Liouville Theorem. Also, this playlist will help: youtube.com/playlist?list=PLdgVBOaXkb9ATVsK2Q84ghjBgIk5faHNc
Sturm-Liouville Video: youtube.com/watch?v=_F0ck1JncLE
Lecture Notes: drive.google.com/file/d/0BzC45hep01Q4cW5WU1ZkeHBqTnc/view?usp=sharing&resourcekey=0-XJHxLDAU_iJZEIJuLqt1DQ
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan
In this video, I describe how to use Green's functions (i.e. responses to single impulse inputs to an ODE) to solve a non-homogeneous (Sturm-Liouville) ODE subject to ANY arbitrary input f(x). All this involves is integrating the Green's function with the input f(x) over the domain. The input, by the way, is the function that makes the ODE non-homogeneous.
This approach is made possible because of the orthogonality relation of the Sturm-Liouville Theorem and the fact that the eigenfunctions of a REGULAR Sturm-Liouville problem form a complete set.
By the way, at 2:05, I say that we can make the eigenvalue 'any real number we want'. I should clarify that and say that we can make the 'n' in lambda_n 'whatever integer we want'. Lambda is still restricted by the boundary conditions. Nonetheless, the point that a smaller set of functions (eigenfunctions of a REGULAR Sturm-Liouville Problem) can be used to describe a larger set of functions (all nice/smooth functions) still holds.
Questions/requests? Let me know in the comments! The oft-requested PDE version of my Green's Function video is coming out soon, so stay tuned!
Prereqs: Know the Sturm-Liouville Theorem. Also, this playlist will help: youtube.com/playlist?list=PLdgVBOaXkb9ATVsK2Q84ghjBgIk5faHNc
Sturm-Liouville Video: youtube.com/watch?v=_F0ck1JncLE
Lecture Notes: drive.google.com/file/d/0BzC45hep01Q4cW5WU1ZkeHBqTnc/view?usp=sharing&resourcekey=0-XJHxLDAU_iJZEIJuLqt1DQ
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan
