59.1.661 problem 678

Internal problem ID [9833]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 678
Date solved : Sunday, March 30, 2025 at 02:47:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

ode:=x^2*diff(diff(y(x),x),x)-(2*5^(1/2)-1)*x*diff(y(x),x)+(19/4-3*x^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{-\frac {1}{2}+\sqrt {5}} \left (c_1 \sinh \left (\sqrt {3}\, x \right )+c_2 \cosh \left (\sqrt {3}\, x \right )\right ) \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 53

ode=x^2*D[y[x],{x,2}]-(2*Sqrt[5]-1)*x*D[y[x],x]+(19/4-3*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{6} e^{-\sqrt {3} x} x^{\sqrt {5}-\frac {1}{2}} \left (\sqrt {3} c_2 e^{2 \sqrt {3} x}+6 c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(-1 + 2*sqrt(5))*Derivative(y(x), x) + (19/4 - 3*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : invalid input: 1 - (-21 + 4*sqrt(5))/(-1 + 2*sqrt(5))