54.3.325 problem 1342

Internal problem ID [12620]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1342
Date solved : Wednesday, October 01, 2025 at 02:18:26 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }&=-\frac {a y}{x^{4}} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 31

ode:=diff(diff(y(x),x),x) = -a/x^4*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 56

ode=D[y[x],{x,2}] == -((a*y[x])/x^4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 x e^{-1+\frac {i \sqrt {a}}{x}}-\frac {i c_2 x e^{1-\frac {i \sqrt {a}}{x}}}{2 \sqrt {a}} \end{align*}

✓ Sympy. Time used: 0.053 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)/x**4 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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