59.3.6 problem Kovacic 1985 paper. page 25. section 5.2. Example 2

Internal problem ID [10008]
Book : Collection of Kovacic problems
Section : section 3. Problems from Kovacic related papers
Problem number : Kovacic 1985 paper. page 25. section 5.2. Example 2
Date solved : Wednesday, March 05, 2025 at 08:02:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x) = -1/36*(5*x^2+27)/(x^2-1)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x^{2}-1}\, \left (\operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, x\right ) c_{1} +\operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, x\right ) c_{2} \right ) \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 38

ode=D[y[x],{x,2}]== -(5*x^2+27)/(36*(x^2-1)^2)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.335 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (5*x**2 + 27)*y(x)/(36*(x**2 - 1)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt [3]{x^{2} - 1} \left (C_{1} \sqrt {x^{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{12}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {x^{2}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{12}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{2}} \right )}\right ) \sqrt [4]{x^{2}}}{\sqrt {x}} \]