54.3.426 problem 1447

Internal problem ID [12721]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1447
Date solved : Wednesday, October 01, 2025 at 02:21:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {y^{\prime }}{x}-\frac {\left (-x -1\right ) y}{x^{4}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x) = -1/x*diff(y(x),x)-(-x-1)/x^4*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 29

ode=D[y[x],{x,2}] == -(((-1 - x)*y[x])/x^4) - D[y[x],x]/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{\frac {1}{x}-\frac {1}{2}} \left (c_1-e c_2 \operatorname {ExpIntegralEi}\left (-\frac {2}{x}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + Derivative(y(x), x)/x + (-x - 1)*y(x)/x**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**4*Derivative(y(x), (x, 2)) + x*y(x) + y(x))/x**3 cannot be solved by the factorable group method