Internal problem ID [8791]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1211.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+4 x^{3} y^{\prime }+\left (4 x^{4}+2 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 41
dsolve(x^2*diff(diff(y(x),x),x)+4*x^3*diff(y(x),x)+(4*x^4+2*x^2+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} {\mathrm e}^{-x^{2}}+c_{2} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} {\mathrm e}^{-x^{2}} \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 60
DSolve[(1 + 2*x^2 + 4*x^4)*y[x] + 4*x^3*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} e^{-x^2} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (3 c_1-i \sqrt {3} c_2 x^{i \sqrt {3}}\right ) \\ \end{align*}