Internal problem ID [8753]
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]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+2 \left (x +a \right ) y^{\prime }-b \left (b -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (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)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\frac {a}{x}} \BesselI \left (b -\frac {1}{2}, \frac {a}{x}\right )}{\sqrt {x}}+\frac {c_{2} {\mathrm e}^{\frac {a}{x}} \BesselK \left (b -\frac {1}{2}, \frac {a}{x}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.09 (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} \, _1F_1\left (1-b;2-2 b;\frac {2 a}{x}\right )+(-2)^b c_2 a^b \left (\frac {1}{x}\right )^b \, _1F_1\left (b;2 b;\frac {2 a}{x}\right ) \\ \end{align*}