54.3.384 problem 1401

Internal problem ID [12679]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1401
Date solved : Wednesday, October 01, 2025 at 02:19:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (3 x^{2}+a \right ) y^{\prime }}{x^{3}}-\frac {b y}{x^{6}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 43

ode:=diff(diff(y(x),x),x) = -1/x^3*(3*x^2+a)*diff(y(x),x)-b/x^6*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_2 \,{\mathrm e}^{\frac {\sqrt {a^{2}-4 b}}{2 x^{2}}}+c_1 \right ) {\mathrm e}^{-\frac {-a +\sqrt {a^{2}-4 b}}{4 x^{2}}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 56

ode=D[y[x],{x,2}] == -((b*y[x])/x^6) - ((a + 3*x^2)*D[y[x],x])/x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{\frac {a-\sqrt {a^2-4 b}}{4 x^2}} \left (c_1 e^{\frac {\sqrt {a^2-4 b}}{2 x^2}}+c_2\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(b*y(x)/x**6 + Derivative(y(x), (x, 2)) + (a + 3*x**2)*Derivative(y(x), x)/x**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-b*y(x) - x**6*Derivative(y(x), (x, 2)))/(x**3*(a + 3*x**2)) cannot be solved by the factorable group method