Internal problem ID [8886]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1307.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 38
dsolve(x^3*diff(diff(y(x),x),x)+x*(x+1)*diff(y(x),x)-2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\frac {1}{x}} \left (x +1\right )}{x}+\frac {c_{2} \left (-{\mathrm e}^{\frac {1}{x}} \left (x +1\right ) \expIntegral \left (1, \frac {1}{x}\right )+x \right )}{x} \]
✓ Solution by Mathematica
Time used: 0.085 (sec). Leaf size: 35
DSolve[-2*y[x] + x*(1 + x)*y'[x] + x^3*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\frac {1}{x}} (x+1) \left (c_1-c_2 \text {Ei}\left (-\frac {1}{x}\right )\right )}{x}-c_2 \\ \end{align*}