58.22.8 problem 4(a)

Internal problem ID [14954]
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 : 4(a)
Date solved : Thursday, October 02, 2025 at 09:56:57 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 18

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

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

✓ Sympy. Time used: 0.098 (sec). Leaf size: 15

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

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