23.3.493 problem 499

Internal problem ID [6207]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 499
Date solved : Tuesday, September 30, 2025 at 02:36:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 33

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

\[ y = c_1 \sinh \left (\sqrt {a}\, \sqrt {x^{2}-1}\right )+c_2 \cosh \left (\sqrt {a}\, \sqrt {x^{2}-1}\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to c_1 \cosh \left (\sqrt {a} \sqrt {x^2-1}\right )+i c_2 \sinh \left (\sqrt {a} \sqrt {x^2-1}\right ) \end{align*}

✗ Sympy

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

False