57.11.1 problem 1(a)

Internal problem ID [14435]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.1 Cauchy-Euler equations. Exercises page 120
Problem number : 1(a)
Date solved : Thursday, October 02, 2025 at 09:37:10 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{\prime \prime }&=-\frac {x}{t^{2}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 29

ode:=diff(diff(x(t),t),t) = -1/t^2*x(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \sqrt {t}\, \left (c_1 \sin \left (\frac {\sqrt {3}\, \ln \left (t \right )}{2}\right )+c_2 \cos \left (\frac {\sqrt {3}\, \ln \left (t \right )}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 42

ode=D[x[t],{t,2}]==-1/t^2*x[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \sqrt {t} \left (c_1 \cos \left (\frac {1}{2} \sqrt {3} \log (t)\right )+c_2 \sin \left (\frac {1}{2} \sqrt {3} \log (t)\right )\right ) \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 34

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

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