59.1.682 problem 699

Internal problem ID [9854]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 699
Date solved : Wednesday, March 05, 2025 at 08:00:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime }+x y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 50

ode:=x^4*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.317 (sec). Leaf size: 57

ode=x^4*D[y[x],{x,2}]+x*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{3/2} \left (x^2-1\right ) \left (c_2 \int _1^x\frac {e^{\frac {1}{2 K[1]^2}-3} K[1]^2}{\left (K[1]^2-1\right )^2}dK[1]+c_1\right )}{x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + y(x),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))/x cannot be solved by the factorable group method