54.3.380 problem 1397

Internal problem ID [12675]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1397
Date solved : Wednesday, October 01, 2025 at 02:19:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 27

ode:=diff(diff(y(x),x),x) = -1/x^4*diff(y(x),x)+1/x^5*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (-3 c_2 \Gamma \left (\frac {2}{3}\right ) \Gamma \left (\frac {1}{3}, -\frac {1}{3 x^{3}}\right )+2 \pi c_2 \sqrt {3}+c_1 \right ) \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 38

ode=D[y[x],{x,2}] == y[x]/x^5 - D[y[x],x]/x^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_2 \Gamma \left (\frac {1}{3},-\frac {1}{3 x^3}\right )}{3^{2/3} \sqrt [3]{-\frac {1}{x^3}}}+c_1 x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + Derivative(y(x), x)/x**4 - y(x)/x**5,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**5*Derivative(y(x), (x, 2)) + y(x))/x cannot be solved by the factorable group method