Internal problem ID [8984]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1405.
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 {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 47
dsolve(diff(diff(y(x),x),x) = (2*x^2+1)/x^3*diff(y(x),x)-1/4*(a*x^4+10*x^2+1)/x^6*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{\frac {3}{2}+\frac {\sqrt {-a +9}}{2}} {\mathrm e}^{-\frac {1}{4 x^{2}}}+c_{2} x^{\frac {3}{2}-\frac {\sqrt {-a +9}}{2}} {\mathrm e}^{-\frac {1}{4 x^{2}}} \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 70
DSolve[y''[x] == -1/4*((1 + 10*x^2 + a*x^4)*y[x])/x^6 + ((1 + 2*x^2)*y'[x])/x^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {1}{4 x^2}} x^{\frac {3}{2}-\frac {\sqrt {9-a}}{2}} \left (c_2 x^{\sqrt {9-a}}+\sqrt {9-a} c_1\right )}{\sqrt {9-a}} \\ \end{align*}