Internal problem ID [7302]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 582.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {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 {gather*}
✓ Solution by Maple
Time used: 0.672 (sec). Leaf size: 107
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)
\[ y \relax (x ) = \frac {c_{1} x^{\frac {1}{3}}}{\left (x^{2}+3\right )^{\frac {2}{3}}}+\frac {c_{2} x^{\frac {1}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}} \sqrt {3}}{6+\left (9 x^{2}+27\right )^{\frac {1}{3}}}\right )-\ln \left (1+\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}}}{3}+\frac {\left (9 x^{2}+27\right )^{\frac {2}{3}}}{9}\right )+2 \ln \left (1-\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}}}{3}\right )\right )}{\left (9 x^{2}+27\right )^{\frac {2}{3}}} \]
✓ Solution by Mathematica
Time used: 0.1 (sec). Leaf size: 94
DSolve[3*x^2*(3+x^2)*y'[x]+x*(3+11*x^2)*y'[x]+(1+5*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \exp \left (\frac {1}{3} \text {RootSum}\left [3 \text {$\#$1}^3+11 \text {$\#$1}^2+9 \text {$\#$1}+3\&,\frac {3 \text {$\#$1}^2 \log (x-\text {$\#$1})-4 \text {$\#$1} \log (x-\text {$\#$1})+9 \log (x-\text {$\#$1})}{9 \text {$\#$1}^2+22 \text {$\#$1}+9}\&\right ]\right )}{\sqrt [3]{x}} \\ y(x)\to 0 \\ \end{align*}