59.1.123 problem 125

Internal problem ID [9295]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 125
Date solved : Monday, January 27, 2025 at 06:01:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 42

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

\[ y = \frac {x \left (\sqrt {2}\, c_{2} \sqrt {x^{2}-2}+2 \arctan \left (\frac {\sqrt {2}}{\sqrt {x^{2}-2}}\right ) c_{2} +c_{1} \right )}{\left (x^{2}-2\right )^{{3}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.252 (sec). Leaf size: 97

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

\[ y(x)\to \exp \left (\int _1^x\frac {1}{2 K[1]-K[1]^3}dK[1]-\frac {1}{2} \int _1^x\left (\frac {5 K[2]}{K[2]^2-2}-\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {1}{2 K[1]-K[1]^3}dK[1]\right )dK[3]+c_1\right ) \]