Internal problem ID [9502]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1167.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (a \,x^{m}+b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 63
dsolve(x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+(a*x^m+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x \operatorname {BesselJ}\left (\frac {2 \sqrt {1-b}}{m}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}}}{m}\right )+c_{2} x \operatorname {BesselY}\left (\frac {2 \sqrt {1-b}}{m}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}}}{m}\right ) \]
✓ Solution by Mathematica
Time used: 0.144 (sec). Leaf size: 130
DSolve[(b + a*x^m)*y[x] - x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to m^{-2/m} a^{\frac {1}{m}} \left (x^m\right )^{\frac {1}{m}} \left (c_1 \operatorname {Gamma}\left (1-\frac {2 i \sqrt {b-1}}{m}\right ) \operatorname {BesselJ}\left (-\frac {2 i \sqrt {b-1}}{m},\frac {2 \sqrt {a} \sqrt {x^m}}{m}\right )+c_2 \operatorname {Gamma}\left (\frac {2 i \sqrt {b-1}}{m}+1\right ) \operatorname {BesselJ}\left (\frac {2 i \sqrt {b-1}}{m},\frac {2 \sqrt {a} \sqrt {x^m}}{m}\right )\right ) \]