59.1.99 problem 101

Internal problem ID [9271]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 101
Date solved : Monday, January 27, 2025 at 06:00:49 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*}

✓ Solution by Maple

Time used: 0.111 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

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

DSolve[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,y[x],x,IncludeSingularSolutions -> True]
 

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