Internal problem ID [8822]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1242.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-\left (3 x +1\right ) y^{\prime }-\left (x^{2}-x \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 42
dsolve((x^2-1)*diff(diff(y(x),x),x)-(3*x+1)*diff(y(x),x)-(x^2-x)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-x} \left (x +1\right )^{2}+c_{2} \left ({\mathrm e}^{-x -2} \left (x +1\right )^{2} \expIntegral \left (1, -2 x -2\right )+2 \,{\mathrm e}^{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.207 (sec). Leaf size: 42
DSolve[(x - x^2)*y[x] - (1 + 3*x)*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x-2} (x+1)^2 \left (c_2 \text {Ei}(2 (x+1))+e^2 c_1\right )-2 c_2 e^x \\ \end{align*}