Internal problem ID [6875]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 143.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+2 y^{\prime } x^{3}+\left (3 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 34
dsolve(4*x^2*diff(y(x),x$2)+2*x^3*diff(y(x),x)+(1+3*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}\, {\mathrm e}^{-\frac {x^{2}}{4}}+c_{2} \sqrt {x}\, {\mathrm e}^{-\frac {x^{2}}{4}} \expIntegral \left (1, -\frac {x^{2}}{4}\right ) \]
✓ Solution by Mathematica
Time used: 0.087 (sec). Leaf size: 39
DSolve[4*x^2*y''[x]+2*x^3*y'[x]+(1+3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-\frac {x^2}{4}} \sqrt {x} \left (c_2 \text {Ei}\left (\frac {x^2}{4}\right )+2 c_1\right ) \\ \end{align*}