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