59.1.306 problem 310

Internal problem ID [9478]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 310
Date solved : Monday, January 27, 2025 at 06:03:15 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 55

dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+12*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{2} \left (5 x^{3}-3 x \right ) \ln \left (x -1\right )}{24}+\frac {\left (-5 x^{3}+3 x \right ) c_{2} \ln \left (x +1\right )}{24}-\frac {5 c_{1} x^{3}}{3}+\frac {5 c_{2} x^{2}}{12}+c_{1} x -\frac {c_{2}}{9} \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 59

DSolve[(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+12*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} c_1 x \left (5 x^2-3\right )+c_2 \left (-\frac {5 x^2}{2}-\frac {1}{4} \left (5 x^2-3\right ) x (\log (1-x)-\log (x+1))+\frac {2}{3}\right ) \]