60.3.338 problem 1344

Internal problem ID [11348]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1344
Date solved : Tuesday, January 28, 2025 at 06:04:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.228 (sec). Leaf size: 23

dsolve(diff(diff(y(x),x),x) = -(exp(2/x)-v^2)/x^4*y(x),y(x), singsol=all)
 

\[ y = x \left (c_{1} \operatorname {BesselJ}\left (v , {\mathrm e}^{\frac {1}{x}}\right )+c_{2} \operatorname {BesselY}\left (v , {\mathrm e}^{\frac {1}{x}}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.504 (sec). Leaf size: 100

DSolve[D[y[x],{x,2}] == -(((E^(2/x) - v^2)*y[x])/x^4),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {(-1)^{-v} 2^{\frac {3 v}{2}+\frac {1}{2}} \left (-e^{2/x}\right )^{-v/2} \left (e^{2/x}\right )^{v/2} \left (c_1 (-1)^v \operatorname {BesselI}\left (v,\sqrt {-e^{2/x}}\right )+c_2 K_v\left (\sqrt {-e^{2/x}}\right )\right )}{\log \left (e^{2/x}\right )} \]