54.3.328 problem 1345

Internal problem ID [12623]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1345
Date solved : Wednesday, October 01, 2025 at 02:18:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 45

ode=D[y[x],{x,2}] == (2*y[x])/x^4 - 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}} x \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)) + 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))/x cannot be solved by the factorable group method