59.1.187 problem 189

Internal problem ID [9359]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 189
Date solved : Monday, January 27, 2025 at 06:01:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 28

dsolve(x^2*(1+x^2)*diff(y(x),x$2)+x*(5+11*x^2)*diff(y(x),x)+24*x^2*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.223 (sec). Leaf size: 112

DSolve[x^2*(1+x^2)*D[y[x],{x,2}]+x*(5+11*x^2)*D[y[x],x]+24*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {3 K[1]^2+5}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {11 K[2]^2+5}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3 K[1]^2+5}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]