Internal problem ID [8022]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 546.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve(6*x^2*diff(y(x),x$2)+x*(1+6*x^2)*diff(y(x),x)+(1+9*x^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x^{\frac {5}{6}}}d x \right ) \]
✓ Solution by Mathematica
Time used: 0.094 (sec). Leaf size: 61
DSolve[6*x^2*y''[x]+x*(1+6*x^2)*y'[x]+(1+9*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {x^2}{2}} \left (2 c_1 x^{11/6}+\sqrt [12]{2} c_2 \left (-x^2\right )^{11/12} \Gamma \left (\frac {1}{12},-\frac {x^2}{2}\right )\right )}{2 x^{3/2}} \]