Chapter 12.15, Problem 5P

Using equations (15.4) and (15.5), show that $J_0(x)=J_2(x)$ at every maximum or minimum of $J_1(x),$ and $J_0(x)=-J_2(x)=J_1^{\prime }(x)$ at every positive zero of $J_1(x)$.

Computer plot $J_0(x),J_1(x),$ and $J_2(x)$ on the same axes, and verify that these results are true. $(15.4)J_{p-1}(x)-J_{p+1}(x)=2J_p^{\prime }(x),$ $(15.5)J_p^{\prime }(x)=-\frac{p}{x}{J}_p(x)+J_{p-1}(x)=\frac{p}{x}{J}_p(x)-J_{p+1}(x).$