Internal problem ID [8750]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1172.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+a y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 57
dsolve(x^2*diff(diff(y(x),x),x)+2*(x-1)*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {1}{x}} \sqrt {\frac {1}{x}}\, \operatorname {BesselI}\left (\frac {\sqrt {-4 a +1}}{2}, \frac {1}{x}\right )+c_{2} {\mathrm e}^{-\frac {1}{x}} \sqrt {\frac {1}{x}}\, \operatorname {BesselK}\left (\frac {\sqrt {-4 a +1}}{2}, \frac {1}{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 121
DSolve[a*y[x] + 2*(-1 + x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{\frac {1}{2}-\sqrt {1-4 a}} e^{-1/x} \sqrt {\frac {1}{x}} \left (c_1 \operatorname {Gamma}\left (1-\frac {1}{2} \sqrt {1-4 a}\right ) \operatorname {BesselI}\left (-\frac {1}{2} \sqrt {1-4 a},\frac {1}{x}\right )+4^{\sqrt {1-4 a}} c_2 \operatorname {Gamma}\left (\frac {1}{2} \sqrt {1-4 a}+1\right ) \operatorname {BesselI}\left (\frac {1}{2} \sqrt {1-4 a},\frac {1}{x}\right )\right ) \\ \end{align*}