59.1.108 problem 110

Internal problem ID [9280]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 110
Date solved : Monday, January 27, 2025 at 06:00:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.100 (sec). Leaf size: 31

dsolve(2*x^2*(1+x^2)*diff(y(x),x$2)+x*(3+8*x^2)*diff(y(x),x)-(3-4*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} x \operatorname {hypergeom}\left (\left [1, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right )+\frac {c_{2}}{\left (x^{2}+1\right )^{{1}/{4}} x^{{3}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.188 (sec). Leaf size: 99

DSolve[2*x^2*(1+x^2)*D[y[x],{x,2}]+x*(3+8*x^2)*D[y[x],x]-(3-4*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x-\frac {3}{4 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {8 K[2]^2+3}{2 \left (K[2]^3+K[2]\right )}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {3}{4 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]