60.3.394 problem 1400

Internal problem ID [11404]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1400
Date solved : Monday, January 27, 2025 at 11:19:32 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }&=\frac {y^{\prime }}{x}-\frac {a y}{x^{6}} \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 35

dsolve(diff(diff(y(x),x),x) = 1/x*diff(y(x),x)-a/x^6*y(x),y(x), singsol=all)
 

\[ y = x^{2} \left (c_{1} \sinh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )+c_{2} \cosh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.145 (sec). Leaf size: 58

DSolve[D[y[x],{x,2}] == -((a*y[x])/x^6) + D[y[x],x]/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} x^2 e^{-\frac {i \sqrt {a}}{2 x^2}} \left (2 c_1 e^{\frac {i \sqrt {a}}{x^2}}-\frac {i c_2}{\sqrt {a}}\right ) \]