In the previous exercise (Section 11.16, Exercise #10) we defined the Bessel functions of the first kind of orders zero and one by,
Prove that these Bessel functions are solutions of the differential equation
when and , respectively.
Proof. In the previous exercise (linked above) we proved the following
For the case we have the differential equation
Plugging in we then have
So, is indeed a solution in the case .
Now, from the previous exercise we have the relations
Starting with the case we differentiate,
Using the relations above from the previous problem, we substitute
Hence, is indeed a solution of the differentiation equation