54.3.383 problem 1400

Internal problem ID [12678]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1400
Date solved : Wednesday, October 01, 2025 at 02:19:31 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 35

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

\[ y = x^{2} \left (c_1 \sinh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )+c_2 \cosh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )\right ) \]

✓ Mathematica. Time used: 0.093 (sec). Leaf size: 58

ode=D[y[x],{x,2}] == -((a*y[x])/x^6) + D[y[x],x]/x; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)/x**6 + Derivative(y(x), (x, 2)) - Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \left (\frac {C_{1} \sqrt {\frac {\sqrt {a}}{x^{2}}} J_{- \frac {1}{2}}\left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {- \frac {\sqrt {a}}{x^{2}}}} + C_{2} Y_{- \frac {1}{2}}\left (- \frac {\sqrt {a}}{2 x^{2}}\right )\right ) \]