87.14.8 problem 8

Internal problem ID [23527]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:42:43 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {9 y}{x^{4}}&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\cos \left (\frac {3}{x}\right ) \end{align*}

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

ode:=diff(diff(y(x),x),x)+2/x*diff(y(x),x)+9/x^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sin \left (\frac {3}{x}\right )+c_2 \cos \left (\frac {3}{x}\right ) \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 25

ode=D[y[x],{x,2}]+2/x*D[y[x],x]+9/x^4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cos \left (\frac {3}{x}\right )-c_2 \sin \left (\frac {3}{x}\right ) \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x)/x + 9*y(x)/x**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\frac {C_{1} \sqrt {\frac {1}{x}} J_{- \frac {1}{2}}\left (\frac {3}{x}\right )}{\sqrt {- \frac {1}{x}}} + C_{2} Y_{- \frac {1}{2}}\left (- \frac {3}{x}\right )}{\sqrt {x}} \]