59.1.107 problem 109

Internal problem ID [9279]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 109
Date solved : Monday, January 27, 2025 at 06:00:56 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 y x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 32

dsolve(x*(1+x^2)*diff(y(x),x$2)+(4+7*x^2)*diff(y(x),x)+8*x*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {-\sqrt {x^{2}+1}\, c_{2} x +\operatorname {arcsinh}\left (x \right ) c_{2} +c_{1}}{\sqrt {x^{2}+1}\, x^{3}} \]

✓ Solution by Mathematica

Time used: 0.220 (sec). Leaf size: 108

DSolve[x*(1+x^2)*D[y[x],{x,2}]+(4+7*x^2)*D[y[x],x]+8*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x-\frac {K[1]^2+2}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {7 K[2]^2+4}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {K[1]^2+2}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]