59.1.566 problem 582

Internal problem ID [9738]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 582
Date solved : Monday, January 27, 2025 at 06:13:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.243 (sec). Leaf size: 102

dsolve(3*x^2*(3+x^2)*diff(y(x),x$2)+x*(3+11*x^2)*diff(y(x),x)+(1+5*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {x^{{1}/{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (9 x^{2}+27\right )^{{1}/{3}} \sqrt {3}}{6+\left (9 x^{2}+27\right )^{{1}/{3}}}\right ) c_{2} +3 \,3^{{1}/{3}} c_{1} +2 \ln \left (3-\left (9 x^{2}+27\right )^{{1}/{3}}\right ) c_{2} -\ln \left (9+3 \left (9 x^{2}+27\right )^{{1}/{3}}+\left (9 x^{2}+27\right )^{{2}/{3}}\right ) c_{2} \right ) 3^{{2}/{3}}}{9 \left (x^{2}+3\right )^{{2}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.104 (sec). Leaf size: 57

DSolve[3*x^2*(3+x^2)*D[y[x],x]+x*(3+11*x^2)*D[y[x],x]+(1+5*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 \exp \left (\int _1^x-\frac {5 K[1]^2+1}{3 K[1]^4+11 K[1]^3+9 K[1]^2+3 K[1]}dK[1]\right ) \\ y(x)\to 0 \\ \end{align*}