59.1.522 problem 538

Internal problem ID [9694]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 538
Date solved : Monday, January 27, 2025 at 06:13:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.107 (sec). Leaf size: 44

dsolve(4*x^2*(1-x^2)*diff(y(x),x$2)+x*(7-19*x^2)*diff(y(x),x)-(1+14*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} \operatorname {LegendreP}\left (-\frac {3}{8}, \frac {5}{8}, \sqrt {-x^{2}+1}\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {3}{8}, \frac {5}{8}, \sqrt {-x^{2}+1}\right )}{x^{{3}/{8}} \sqrt {x^{2}-1}} \]

✓ Solution by Mathematica

Time used: 0.291 (sec). Leaf size: 120

DSolve[4*x^2*(1-x^2)*D[y[x],{x,2}]+x*(7-19*x^2)*D[y[x],x]-(1+14*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x-\frac {3 K[1]^2+1}{8 K[1]-8 K[1]^3}dK[1]-\frac {1}{2} \int _1^x\frac {7-19 K[2]^2}{4 K[2]-4 K[2]^3}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {3 K[1]^2+1}{8 K[1]-8 K[1]^3}dK[1]\right )dK[3]+c_1\right ) \]