Internal problem ID [8923]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1344.
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 ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.028 (sec). Leaf size: 23
dsolve(diff(diff(y(x),x),x) = -(exp(2/x)-v^2)/x^4*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \BesselJ \left (v , {\mathrm e}^{\frac {1}{x}}\right )+c_{2} x \BesselY \left (v , {\mathrm e}^{\frac {1}{x}}\right ) \]
✓ Solution by Mathematica
Time used: 0.329 (sec). Leaf size: 100
DSolve[y''[x] == -(((E^(2/x) - v^2)*y[x])/x^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(-1)^{-v} 2^{\frac {3 v}{2}+\frac {1}{2}} \left (-e^{2/x}\right )^{-v/2} \left (e^{2/x}\right )^{v/2} \left (c_1 (-1)^v I_v\left (\sqrt {-e^{2/x}}\right )+c_2 K_v\left (\sqrt {-e^{2/x}}\right )\right )}{\log \left (e^{2/x}\right )} \\ \end{align*}