Internal problem ID [10947]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 113.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+n \left (n +1\right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 41
dsolve(x^2*diff(y(x),x$2)-(a^2*x^2+n*(n+1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (n +\frac {1}{2}, \sqrt {-a^{2}}\, x \right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (n +\frac {1}{2}, \sqrt {-a^{2}}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 42
DSolve[x^2*y''[x]-(a^2*x^2+n*(n+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (n+\frac {1}{2},-i a x\right )+c_2 \operatorname {BesselY}\left (n+\frac {1}{2},-i a x\right )\right ) \]