54.3.337 problem 1354

Internal problem ID [12632]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1354
Date solved : Wednesday, October 01, 2025 at 02:18:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}} \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x) = 1/x^3*(2*x^2-1)*diff(y(x),x)-2/x^4*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2 \,x^{5} \operatorname {hypergeom}\left (\left [-\frac {5}{2}\right ], \left [-\frac {1}{2}\right ], \frac {1}{2 x^{2}}\right )+5 c_1 \,x^{2}-c_1}{x^{2}} \]

✓ Mathematica. Time used: 0.275 (sec). Leaf size: 65

ode=D[y[x],{x,2}] == (-2*y[x])/x^4 + ((-1 + 2*x^2)*D[y[x],x])/x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{9/2} \left (5 x^2-1\right ) \left (c_2 \int _1^x\frac {25 e^{\frac {1}{2 K[1]^2}-9} K[1]^6}{\left (1-5 K[1]^2\right )^2}dK[1]+c_1\right )}{5 x^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - (2*x**2 - 1)*Derivative(y(x), x)/x**3 + 2*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)) + 2*y(x))/(2*x**3 - x) cannot be solved by the factorable group method