59.1.609 problem 625

Internal problem ID [9781]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 625
Date solved : Monday, January 27, 2025 at 06:14:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 29

dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(5-x^2)*diff(y(x),x)-(7+25*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {4 c_{2} x^{10}+5 c_{2} x^{8}+c_{1}}{x \left (x^{2}+1\right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.200 (sec). Leaf size: 110

DSolve[x^2*(1+x^2)*D[y[x],{x,2}]-x*(5-x^2)*D[y[x],x]-(7+25*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x-\frac {9 K[1]^2+7}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {K[2]^2-5}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {9 K[1]^2+7}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]