60.3.332 problem 1349

Internal problem ID [11328]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1349
Date solved : Sunday, March 30, 2025 at 08:17:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 65

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

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

✓ Mathematica. Time used: 0.216 (sec). Leaf size: 73

ode=D[y[x],{x,2}] == -(y[x]/x^4) - ((1 + x^2)*D[y[x],x])/x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {1}{2 x^2}| \begin {array}{c} \frac {3}{2} \\ 0,0 \\ \end {array} \right )+\frac {c_1 e^{\frac {1}{4 x^2}} \left (\left (2 x^2-1\right ) \operatorname {BesselI}\left (0,\frac {1}{4 x^2}\right )+\operatorname {BesselI}\left (1,\frac {1}{4 x^2}\right )\right )}{2 x^2} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 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))/(x**3 + x) cannot be solved by the factorable group method