54.3.334 problem 1351

Internal problem ID [12629]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1351
Date solved : Wednesday, October 01, 2025 at 02:18:33 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: 0.005 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 44

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 {1}{2} e^{\frac {1}{2 x^2}} \left (2 c_1-\sqrt {2 \pi } c_2 \text {erf}\left (\frac {1}{\sqrt {2} x}\right )\right ) \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