59.1.696 problem 713

Internal problem ID [9868]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 713
Date solved : Sunday, March 30, 2025 at 02:48:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 24

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

\[ y = c_1 \left (x^{2}-1\right )^{{3}/{4}}+c_2 \left (x^{2}-1\right )^{{1}/{4}} x \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 51

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

\[ y(x)\to \sqrt {x^2-1} \left (c_2 Q_{\frac {1}{2}}^{\frac {1}{2}}(x)+\frac {\sqrt {\frac {2}{\pi }} c_1 x}{\sqrt [4]{1-x^2}}\right ) \]

✓ Sympy. Time used: 0.324 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*Derivative(y(x), (x, 2)) + (6 - 3*x**2)*y(x)/(1 - x**2)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt [4]{x^{2} - 1} \left (C_{1} \sqrt {x^{2}} {{}_{2}F_{1}\left (\begin {matrix} 0, 1 \\ \frac {3}{2} \end {matrix}\middle | {x^{2}} \right )} + C_{2} {{}_{1}F_{0}\left (\begin {matrix} - \frac {1}{2} \\ \end {matrix}\middle | {x^{2}} \right )}\right ) \sqrt [4]{x^{2}}}{\sqrt {x}} \]