54.3.336 problem 1353

Internal problem ID [12631]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1353
Date solved : Wednesday, October 01, 2025 at 02:18:35 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 {y}{x^{4}} \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 64

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

\[ y = \frac {c_1 \sqrt {2}\, \sqrt {\pi }\, \left (x^{4}+2 x^{2}-1\right ) \operatorname {erfi}\left (\frac {\sqrt {2}}{2 x}\right )+2 c_1 \left (-x^{3}+x \right ) {\mathrm e}^{\frac {1}{2 x^{2}}}+c_2 \left (x^{4}+2 x^{2}-1\right )}{x} \]

✓ Mathematica. Time used: 0.829 (sec). Leaf size: 66

ode=D[y[x],{x,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^3 \left (x^4+2 x^2-1\right ) \left (c_2 \int _1^x\frac {e^{\frac {1}{2 K[1]^2}-6} K[1]^4}{\left (K[1]^4+2 K[1]^2-1\right )^2}dK[1]+c_1\right )}{x} \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 + 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)) + y(x))/(2*x**3 - x) cannot be solved by the factorable group method