Calculating differential equation with bessel function in it with ln

Question

How can I solve that:$$\frac{d}{dx}\ln \left(\frac{xJ_{-\frac14}\left(\frac{x^2}{2}\right)}{2} \right)$$without using the fact$$2J^{'}_{\nu}(z)=J_{\nu-1}(z)-J_{\nu+1}(z)$$only the definition of the Bessel function

Answer 1

Hint: The derivative of a Bessel function satisfies \begin{eqnarray*} 2J^{'}_{\nu}(z)=J_{\nu-1}(z)-J_{\nu+1}(z) \end{eqnarray*} where differentiation is wrt to $z$. You will still need to differentiate a product & some function of a function ... but thats easy ?

Answer 2

Another Hint: $$\beta(x)= \frac{d}{dx}\ln \left(\frac{xJ_{-\frac14}\left(\frac{x^2}{2}\right)}{2} \right)$$ $$\beta(x)=\frac{d}{dx}\ln \left({J_{-\frac14}\left(\frac{x^2}{2}\right)} \right)+\frac {d}{dx} \left ( \ln \frac x2 \right ) $$

Then use for the first derivative your question here : Calculating the integral of some bessel function $$\int \frac {x\left(J_{ \frac 34}(\frac {x^2}{2})-J_{- \frac 54}(\frac {x^2}{2})\right)}{2J_{- \frac 14}(\frac {x^2}{2})}dx=-\ln \left ( J_{- \frac 14}(\frac {x^2}{2}) \right )$$ Take derivative on both sides $$ \frac {x\left(J_{ \frac 34}(\frac {x^2}{2})-J_{- \frac 54}(\frac {x^2}{2})\right)}{2J_{- \frac 14}(\frac {x^2}{2})}=-\frac {d}{dx}\left (\ln \left ( J_{- \frac 14}(\frac {x^2}{2}) \right ) \right )$$