59.1.111 problem 113

Internal problem ID [9283]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 113
Date solved : Monday, January 27, 2025 at 06:00:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.149 (sec). Leaf size: 33

dsolve(9*x^2*(1+x^2)*diff(y(x),x$2)+3*x*(3+13*x^2)*diff(y(x),x)-(1-25*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.239 (sec). Leaf size: 113

DSolve[9*x^2*(1+x^2)*D[y[x],{x,2}]+3*x*(3+13*x^2)*D[y[x],x]-(1-25*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {3 K[1]^2+1}{6 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\left (\frac {10 K[2]}{3 \left (K[2]^2+1\right )}+\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3 K[1]^2+1}{6 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]