60.3.326 problem 1343

Internal problem ID [11322]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1343
Date solved : Thursday, March 13, 2025 at 08:50:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 58

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

\[ y = \operatorname {BesselI}\left (a +1, \frac {b}{x}\right ) c_{1} b -\operatorname {BesselK}\left (a +1, \frac {b}{x}\right ) c_{2} b +2 \left (a x +\frac {b}{2}\right ) \left (c_{1} \operatorname {BesselI}\left (a , \frac {b}{x}\right )+c_{2} \operatorname {BesselK}\left (a , \frac {b}{x}\right )\right ) \]

✓ Mathematica. Time used: 0.253 (sec). Leaf size: 65

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

\[ y(x)\to c_1 (2 a x+b) \operatorname {BesselI}\left (a,\frac {b}{x}\right )+b c_1 \operatorname {BesselI}\left (a+1,\frac {b}{x}\right )+c_2 \left ((2 a x+b) K_a\left (\frac {b}{x}\right )-b K_{a+1}\left (\frac {b}{x}\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (a*x**2*(1 - a) - b*(b + x))*y(x)/x**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve Derivative(y(x), (x, 2)) + (a*x**2*(1 - a) - b*(b + x))*y(x)/x**4