Internal problem ID [8702]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1122.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x -\left (x^{2}-x -2\right ) y^{\prime }-x \left (x +3\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.113 (sec). Leaf size: 34
dsolve(x*diff(diff(y(x),x),x)-(x^2-x-2)*diff(y(x),x)-x*(x+3)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\frac {x^{2}}{2}} c_{1}+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{-\frac {x \left (x +2\right )}{2}}}{x^{2}}d x \right ) \]
✓ Solution by Mathematica
Time used: 0.341 (sec). Leaf size: 45
DSolve[-(x*(3 + x)*y[x]) - (-2 - x + x^2)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {x^2}{2}} \left (c_2 \int _1^x\frac {e^{-\frac {1}{2} K[1] (K[1]+2)}}{K[1]^2}dK[1]+c_1\right ) \\ \end{align*}