58.22.1 problem 1 (a)

Internal problem ID [14947]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 11, The nth order homogeneous linear differential equation. Section 11.8, Exercises page 583
Problem number : 1 (a)
Date solved : Thursday, October 02, 2025 at 09:56:51 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 27

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

\[ x = \frac {c_1 \sin \left (\sqrt {2}\, \ln \left (t \right )\right )+c_2 \cos \left (\sqrt {2}\, \ln \left (t \right )\right )}{t} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 34

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

\begin{align*} x(t)&\to \frac {c_2 \cos \left (\sqrt {2} \log (t)\right )+c_1 \sin \left (\sqrt {2} \log (t)\right )}{t} \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 27

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

\[ x{\left (t \right )} = \frac {C_{1} \sin {\left (\sqrt {2} \log {\left (t \right )} \right )} + C_{2} \cos {\left (\sqrt {2} \log {\left (t \right )} \right )}}{t} \]