59.1.354 problem 361

Internal problem ID [9526]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 361
Date solved : Monday, January 27, 2025 at 06:03:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 0.311 (sec). Leaf size: 81

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

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