59.1.396 problem 408

Internal problem ID [9568]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 408
Date solved : Monday, January 27, 2025 at 06:04:11 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 14

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

\[ y = c_{2} x^{2}+c_{1} x +c_{2} \]

✓ Solution by Mathematica

Time used: 0.351 (sec). Leaf size: 75

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

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