58.13.6 problem 6

Internal problem ID [14817]
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 : 6
Date solved : Thursday, October 02, 2025 at 09:55:17 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-3 x y^{\prime }+13 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=x^2*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+13*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{2} \left (c_1 \sin \left (3 \ln \left (x \right )\right )+c_2 \cos \left (3 \ln \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 26

ode=x^2*D[y[x],{x,2}]-3*x*D[y[x],x]+13*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^2 (c_2 \cos (3 \log (x))+c_1 \sin (3 \log (x))) \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 3*x*Derivative(y(x), x) + 13*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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