54.3.290 problem 1307

Internal problem ID [12585]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1307
Date solved : Wednesday, October 01, 2025 at 02:10:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-2 y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

ode:=x^3*diff(diff(y(x),x),x)+x*(1+x)*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.268 (sec). Leaf size: 51

ode=-2*y[x] + x*(1 + x)*D[y[x],x] + x^3*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) + x*(x + 1)*Derivative(y(x), x) - 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**3*Derivative(y(x), (x, 2)) + 2*y(x))/(x*(x + 1)) cannot be solved by the factorable group method