90.22.8 problem 8

Internal problem ID [25346]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 353
Problem number : 8
Date solved : Friday, October 03, 2025 at 12:00:27 AM
CAS classification : [[_Emden, _Fowler]]

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

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

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

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 42

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

\begin{align*} y(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: 0.057 (sec). Leaf size: 34

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

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