Internal problem ID [8774]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1194.
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 }+x \left (x +1\right ) y^{\prime }+\left (3 x -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 44
dsolve(x^2*diff(diff(y(x),x),x)+x*(x+1)*diff(y(x),x)+(3*x-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \,{\mathrm e}^{-x} \left (x -3\right )+\frac {c_{2} \left ({\mathrm e}^{-x} x^{2} \left (x -3\right ) \expIntegral \left (1, -x \right )+x^{2}-2 x -1\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.06 (sec). Leaf size: 45
DSolve[(-1 + 3*x)*y[x] + x*(1 + x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} e^{-x} (x-3) x (c_2 \text {Ei}(x)+6 c_1)-\frac {c_2 ((x-2) x-1)}{6 x} \\ \end{align*}