Internal problem ID [10959]
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: 125.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -\left (x^{2}+\left (n +\frac {1}{2}\right )^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 19
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)-(x^2+(n+1/2)^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {BesselI}\left (n +\frac {1}{2}, x\right )+c_{2} \operatorname {BesselK}\left (n +\frac {1}{2}, x\right ) \]
✓ Solution by Mathematica
Time used: 0.056 (sec). Leaf size: 34
DSolve[x^2*y''[x]+x*y'[x]-(x^2+(n+1/2)^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {BesselJ}\left (n+\frac {1}{2},-i x\right )+c_2 \operatorname {BesselY}\left (n+\frac {1}{2},-i x\right ) \]