59.1.48 problem 50

Internal problem ID [9220]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 50
Date solved : Monday, January 27, 2025 at 05:52:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 42

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

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

✓ Solution by Mathematica

Time used: 0.788 (sec). Leaf size: 73

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

\[ y(x)\to \frac {\left (c_1 x-c_2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{2},\frac {1}{2},-2 x^2\right )\right ) \exp \left (-\frac {1}{2} \int _1^x\frac {2 \left (K[1]^2+2\right )}{2 K[1]^3+K[1]}dK[1]\right )}{\left (2 x^2+1\right )^{3/4}} \]