Internal problem ID [8769]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1189.
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}+a x y^{\prime }+\left (b \,x^{m}+c \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 85
dsolve(x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(b*x^m+c)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {a}{2}+\frac {1}{2}} \BesselJ \left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right )+c_{2} x^{-\frac {a}{2}+\frac {1}{2}} \BesselY \left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) \]
✓ Solution by Mathematica
Time used: 0.032 (sec). Leaf size: 161
DSolve[(c + b*x^m)*y[x] + a*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to m^{\frac {a-1}{m}} b^{-\frac {a-1}{2 m}} \left (x^m\right )^{-\frac {a-1}{2 m}} \left (c_1 \text {Gamma}\left (\frac {m-\sqrt {(a-1)^2-4 c}}{m}\right ) J_{-\frac {\sqrt {(a-1)^2-4 c}}{m}}\left (\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )+c_2 \text {Gamma}\left (\frac {\sqrt {(a-1)^2-4 c}+m}{m}\right ) J_{\frac {\sqrt {(a-1)^2-4 c}}{m}}\left (\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )\right ) \\ \end{align*}