Internal problem ID [9499]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1169.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x +\left (x a -b^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 49
dsolve(x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+(a*x-b^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \operatorname {BesselJ}\left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.103 (sec). Leaf size: 103
DSolve[(-b^2 + a*x)*y[x] + 2*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_1 \operatorname {Gamma}\left (1-\sqrt {4 b^2+1}\right ) \operatorname {BesselJ}\left (-\sqrt {4 b^2+1},2 \sqrt {a} \sqrt {x}\right )+c_2 \operatorname {Gamma}\left (\sqrt {4 b^2+1}+1\right ) \operatorname {BesselJ}\left (\sqrt {4 b^2+1},2 \sqrt {a} \sqrt {x}\right )}{\sqrt {a} \sqrt {x}} \]