80.6.29 problem 33

Internal problem ID [21319]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 6. Higher order linear equations. Excercise 6.5 at page 133
Problem number : 33
Date solved : Thursday, October 02, 2025 at 07:28:24 PM
CAS classification : [[_3rd_order, _exact, _linear, _homogeneous]]

\begin{align*} t^{3} x^{\prime \prime \prime }+4 t^{2} x^{\prime \prime }+3 x^{\prime } t +x&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

ode:=t^3*diff(diff(diff(x(t),t),t),t)+4*t^2*diff(diff(x(t),t),t)+3*t*diff(x(t),t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {c_1}{t}+c_2 \sin \left (\ln \left (t \right )\right )+c_3 \cos \left (\ln \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 24

ode=t^3*D[x[t],{t,3}]+4*t^2*D[x[t],{t,2}]+3*t*D[x[t],t]+x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {c_3}{t}+c_1 \cos (\log (t))+c_2 \sin (\log (t)) \end{align*}

✓ Sympy. Time used: 0.130 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**3*Derivative(x(t), (t, 3)) + 4*t**2*Derivative(x(t), (t, 2)) + 3*t*Derivative(x(t), t) + x(t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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