54.3.95 problem 1109

Internal problem ID [12390]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1109
Date solved : Wednesday, October 01, 2025 at 01:44:15 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }-x y^{\prime }-y-x \left (x +1\right ) {\mathrm e}^{x}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

ode:=x*diff(diff(y(x),x),x)-x*diff(y(x),x)-y(x)-x*(1+x)*exp(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (-c_1 \,\operatorname {Ei}_{1}\left (x \right ) x +{\mathrm e}^{-x} c_1 -\ln \left (x \right ) x +c_2 x +x^{2}-1\right ) \]

✓ Mathematica. Time used: 0.198 (sec). Leaf size: 85

ode=-(E^x*x*(1 + x)) - y[x] - x*D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x x \left (\left (e^x \left (x^2-x+1\right )+c_2\right ) \int _1^x\frac {e^{-K[1]}}{K[1]^2}dK[1]+\int _1^x-e^{K[2]} K[2] (K[2]+1) \int _1^{K[2]}\frac {e^{-K[1]}}{K[1]^2}dK[1]dK[2]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x + 1)*exp(x) - x*Derivative(y(x), x) + x*Derivative(y(x), (x, 2)) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x*exp(x) + exp(x) + Derivative(y(x), x) - Derivative(y(x), (x, 2)) + y(x)/x cannot be solved by the factorable group method