59.1.191 problem 193

Internal problem ID [9363]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 193
Date solved : Monday, January 27, 2025 at 06:01:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 27

dsolve(4*x^2*(1+x^2)*diff(y(x),x$2)+4*x*(2+x^2)*diff(y(x),x)-(15+x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{2} \left (x^{2}+1\right )^{{3}/{2}}+3 c_{1} x^{2}+2 c_{1}}{x^{{5}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.212 (sec). Leaf size: 110

DSolve[4*x^2*(1+x^2)*D[y[x],{x,2}]+4*x*(2+x^2)*D[y[x],x]-(15+x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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