59.1.166 problem 168

Internal problem ID [9338]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 168
Date solved : Monday, January 27, 2025 at 06:01:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.696 (sec). Leaf size: 59

dsolve(x^2*(1+x^2)*diff(y(x),x$2)+x*(3+10*x^2)*diff(y(x),x)-(15-14*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right ) c_{2} x^{8}-c_{2} \left (x^{4}-\frac {8}{3} x^{2}-\frac {8}{3}\right ) \left (x^{2}+2\right ) \sqrt {x^{2}+1}+c_{1} x^{8}}{\left (x^{2}+1\right )^{{5}/{2}} x^{5}} \]

✓ Solution by Mathematica

Time used: 0.260 (sec). Leaf size: 112

DSolve[x^2*(1+x^2)*D[y[x],{x,2}]+x*(3+10*x^2)*D[y[x],x]-(15-14*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {6 K[1]^2+9}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {10 K[2]^2+3}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {6 K[1]^2+9}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]