The problem asks you to show that \(\displaystyle F(x)= xJ_n(x)\) satisfies the differential equation \(\displaystyle x^2F''(x)- xF'(x)+ [(bx)^2+ 1- n^2]= 0\).
The way you start any problem that asks you to show that a function satisfies a given equation is to put the function into the equation and see what happens! With \(\displaystyle F(x)= xJ_n(x)\), \(\displaystyle F'(x)= J_n(x)+ xJ'_n(x)\) and \(\displaystyle F''(x)= 2J'_n(x)+ xJ''(x)\). Now, do you know what differential equation \(\displaystyle J_n(x)\) satisfies?
