58.13.9 problem 9

Internal problem ID [14820]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.5. The Cauchy-Euler Equation. Exercises page 169
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:55:20 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 9 x^{2} y^{\prime \prime }+3 x y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

ode:=9*x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\ln \left (x \right ) c_2 +c_1 \right ) x^{{1}/{3}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 24

ode=9*x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} \sqrt [3]{x} (c_2 \log (x)+3 c_1) \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt [3]{x} \left (C_{1} + C_{2} \log {\left (x \right )}\right ) \]