Bessel's Equation - Ordinary Differential Equations | Lecture 29

In this lecture we summarize the Frobenius method in complete generality. We further discuss what happens when the indicial equation has repeated roots, which leads to a significantly more complex form for the second fundamental solution to the ODE. We then apply these methods to Bessel's equation, a famous second-order ODE that arises in many areas of applied mathematics. With Bessel's equation we demonstrate the complexity of identifying series solutions when there are repeated roots of the indicial equation.

* In this video we focus on the Bessel equation of 0th order, meaning that the parameter alpha is taken to be 0. You can work with other values of alpha and follow similarly to arrive at the solutions there too.

Two of my recent papers that use the Bessel equation and the functions that solve it (all using the standard notation of J_nu(x) ):

This course is taught by Jason Bramburger for Concordia University.

Follow @jbramburger7 on Twitter for updates.

[ashraharkin]

You are one of my favourite maths teachers on youtube. I love you teaching style and the way your videos are put together. Thank you for all you efforts :))

[nateplus]

Holy Smokes! I completely understand Bessel's Equation now

[AlphaBeta-xtwn]

This has been excellent. Thank you very much.

[ayusuf]

little comment, on 22:50 finding the derivative of ugly product, one doesn't have to do product rule also the same result can be obtained by taking ln of a_2m and distributing (-1)^n to each squared term in denominator and solving those parts. (a_2m)'/(a_2m) is basically (ln(a_2m))' . Maybe this doesn't make sense when a_2m is negative but this isn't our issue.

[larrydurante]

I don't see how at 9:00 you got the double sum into the form you did ?