Internal problem ID [8689]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1109.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-x y^{\prime }-y-x \left (x +1\right ) {\mathrm e}^{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
dsolve(x*diff(diff(y(x),x),x)-x*diff(y(x),x)-y(x)-x*(x+1)*exp(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \left (-\expIntegral \left (1, x\right ) x +{\mathrm e}^{-x}\right ) {\mathrm e}^{x} c_{1}+{\mathrm e}^{x} c_{2} x -\left (x \ln \relax (x )-x^{2}+1\right ) {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.077 (sec). Leaf size: 38
DSolve[-(E^x*x*(1 + x)) - y[x] - x*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^x (-x (\log (-x)+c_2 \text {Ei}(-x))+x (x+1+c_1)-1)-c_2 \\ \end{align*}