82.53.1 problem Ex. 1

Internal problem ID [18942]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at page 118
Problem number : Ex. 1
Date solved : Monday, March 31, 2025 at 06:26:07 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)/x+a^2/x^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sin \left (\frac {a}{x}\right )+c_2 \cos \left (\frac {a}{x}\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 25

ode=D[y[x],{x,2}]+2/x*D[y[x],x]+a^2/x^4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos \left (\frac {a}{x}\right )-c_2 \sin \left (\frac {a}{x}\right ) \]

✓ Sympy. Time used: 0.227 (sec). Leaf size: 44

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

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