Internal problem ID [6810]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 78.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 56
dsolve(diff(y(x),x$2)+3*x*diff(y(x),x)+(4+2*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\sqrt {2}\, {\mathrm e}^{-x^{2}} \sqrt {\pi }\, \left (x -1\right ) \left (x +1\right ) \erfi \left (\frac {x \sqrt {2}}{2}\right )-2 x \,{\mathrm e}^{-\frac {x^{2}}{2}}\right )+c_{2} {\mathrm e}^{-x^{2}} \left (x^{2}-1\right ) \]
✓ Solution by Mathematica
Time used: 0.267 (sec). Leaf size: 60
DSolve[y''[x]+3*x*y'[x]+(4+2*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} e^{-x^2} \left (\left (x^2-1\right ) \left (\sqrt {2 \pi } c_2 \text {Erfi}\left (\frac {x}{\sqrt {2}}\right )+4 c_1\right )-2 c_2 e^{\frac {x^2}{2}} x\right ) \\ \end{align*}