59.1.602 problem 618

Internal problem ID [9774]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 618
Date solved : Monday, January 27, 2025 at 06:14:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 56

dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(7-2*x^2)*diff(y(x),x)+12*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {x^{2} \left (-15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right ) c_{2} x^{4}+c_{2} \left (8 x^{4}-9 x^{2}-2\right ) \sqrt {x^{2}+1}+c_{1} x^{4}\right )}{\left (x^{2}+1\right )^{{7}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.170 (sec). Leaf size: 96

DSolve[x^2*(1+x^2)*D[y[x],{x,2}]-x*(7-2*x^2)*D[y[x],x]+12*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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