Internal problem ID [8932]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1353.
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: 0.016 (sec). Leaf size: 69
dsolve(diff(diff(y(x),x),x) = 1/x^3*(2*x^2-1)*diff(y(x),x)-1/x^4*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (-\sqrt {2}\, \sqrt {\pi }\, \left (x^{4}+2 x^{2}-1\right ) \erfi \left (\frac {\sqrt {2}}{2 x}\right )+\left (2 x^{3}-2 x \right ) {\mathrm e}^{\frac {1}{2 x^{2}}}\right )}{x}+\frac {c_{2} \left (x^{4}+2 x^{2}-1\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.088 (sec). Leaf size: 69
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 {\left (x^4+2 x^2-1\right ) \left (16 c_1-\sqrt {2 \pi } c_2 \text {Erfi}\left (\frac {1}{\sqrt {2} x}\right )\right )+2 c_2 e^{\frac {1}{2 x^2}} x \left (x^2-1\right )}{16 x} \\ \end{align*}