Internal problem ID [8979]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1400.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}+\frac {a y}{x^{6}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x) = 1/x*diff(y(x),x)-a/x^6*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{2} \sinh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )+c_{2} x^{2} \cosh \left (\frac {\sqrt {-a}}{2 x^{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.072 (sec). Leaf size: 58
DSolve[y''[x] == -((a*y[x])/x^6) + y'[x]/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} x^2 e^{-\frac {i \sqrt {a}}{2 x^2}} \left (2 c_1 e^{\frac {i \sqrt {a}}{x^2}}-\frac {i c_2}{\sqrt {a}}\right ) \\ \end{align*}