59.1.99 problem 101

Internal problem ID [9271]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 101
Date solved : Sunday, March 30, 2025 at 02:31:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 34

ode:=8*x^2*(-x^2+1)*diff(diff(y(x),x),x)+2*x*(-13*x^2+1)*diff(y(x),x)+(-9*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{{1}/{4}} \left (\operatorname {LegendreQ}\left (-\frac {1}{8}, \frac {1}{8}, \sqrt {-x^{2}+1}\right ) c_2 \,x^{{1}/{8}}+c_1 \right )}{\sqrt {x^{2}-1}} \]

✓ Mathematica. Time used: 0.308 (sec). Leaf size: 118

ode=8*x^2*(1-x^2)*D[y[x],{x,2}]+2*x*(1-13*x^2)*D[y[x],x]+(1-9*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {3-7 K[1]^2}{8 K[1]-8 K[1]^3}dK[1]-\frac {1}{2} \int _1^x\frac {1-13 K[2]^2}{4 K[2]-4 K[2]^3}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3-7 K[1]^2}{8 K[1]-8 K[1]^3}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*(1 - x**2)*Derivative(y(x), (x, 2)) + 2*x*(1 - 13*x**2)*Derivative(y(x), x) + (1 - 9*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False