Internal problem ID [8747]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}-x y^{\prime }+\left (a \,x^{m}+b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \relax (x ) = c_{1} x \BesselJ \left (\frac {2 \sqrt {-b +1}}{m}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}}}{m}\right )+c_{2} x \BesselY \left (\frac {2 \sqrt {-b +1}}{m}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}}}{m}\right ) \]
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 130
DSolve[(b + a*x^m)*y[x] - x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to m^{-2/m} a^{\frac {1}{m}} \left (x^m\right )^{\frac {1}{m}} \left (c_1 \text {Gamma}\left (1-\frac {2 i \sqrt {b-1}}{m}\right ) J_{-\frac {2 i \sqrt {b-1}}{m}}\left (\frac {2 \sqrt {a} \sqrt {x^m}}{m}\right )+c_2 \text {Gamma}\left (1+\frac {2 i \sqrt {b-1}}{m}\right ) J_{\frac {2 i \sqrt {b-1}}{m}}\left (\frac {2 \sqrt {a} \sqrt {x^m}}{m}\right )\right ) \\ \end{align*}