Internal problem ID [8751]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1173.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+2 \left (a +x \right ) y^{\prime }-b \left (b -1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 45
dsolve(x^2*diff(diff(y(x),x),x)+2*(x+a)*diff(y(x),x)-b*(b-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\frac {a}{x}} \operatorname {BesselI}\left (b -\frac {1}{2}, \frac {a}{x}\right )}{\sqrt {x}}+\frac {c_{2} {\mathrm e}^{\frac {a}{x}} \operatorname {BesselK}\left (b -\frac {1}{2}, \frac {a}{x}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.091 (sec). Leaf size: 74
DSolve[(1 - b)*b*y[x] + 2*(a + x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (-2)^{1-b} c_1 a^{1-b} \left (\frac {1}{x}\right )^{1-b} \operatorname {Hypergeometric1F1}\left (1-b,2-2 b,\frac {2 a}{x}\right )+(-2)^b c_2 a^b \left (\frac {1}{x}\right )^b \operatorname {Hypergeometric1F1}\left (b,2 b,\frac {2 a}{x}\right ) \\ \end{align*}