Internal problem ID [8930]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1351.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 30
dsolve(diff(diff(y(x),x),x) = -(2*x^2+1)/x^3*diff(y(x),x)+1/x^4*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {1}{2 x^{2}}}+c_{2} {\mathrm e}^{\frac {1}{2 x^{2}}} \erf \left (\frac {\sqrt {2}}{2 x}\right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 44
DSolve[y''[x] == y[x]/x^4 - ((1 + 2*x^2)*y'[x])/x^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{\frac {1}{2 x^2}} \left (2 c_1-\sqrt {2 \pi } c_2 \text {Erf}\left (\frac {1}{\sqrt {2} x}\right )\right ) \\ \end{align*}