60.3.292 problem 1309

Internal problem ID [11288]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1309
Date solved : Sunday, March 30, 2025 at 08:08:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+x y&=0 \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 65

ode:=x^3*diff(diff(y(x),x),x)-(x^2-1)*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\frac {1}{4 x^{2}}} \left (c_1 \operatorname {BesselI}\left (0, \frac {1}{4 x^{2}}\right ) \left (2 x^{2}-1\right )+c_2 \operatorname {BesselK}\left (0, -\frac {1}{4 x^{2}}\right ) \left (2 x^{2}-1\right )+\operatorname {BesselI}\left (1, \frac {1}{4 x^{2}}\right ) c_1 +\operatorname {BesselK}\left (1, -\frac {1}{4 x^{2}}\right ) c_2 \right )}{x} \]

✓ Mathematica. Time used: 0.225 (sec). Leaf size: 77

ode=x*y[x] - (-1 + x^2)*D[y[x],x] + x^3*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {1}{2 x^2}| \begin {array}{c} 1 \\ -\frac {1}{2},-\frac {1}{2} \\ \end {array} \right )+\frac {c_1 e^{\frac {1}{4 x^2}} \left (\left (2 x^2-1\right ) \operatorname {BesselI}\left (0,\frac {1}{4 x^2}\right )+\operatorname {BesselI}\left (1,\frac {1}{4 x^2}\right )\right )}{\sqrt {2} x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) + x*y(x) - (x**2 - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(x**2*Derivative(y(x), (x, 2)) + y(x))/(x**2 - 1) + Derivative(y(x), x) cannot be solved by the factorable group method