90.22.10 problem 10

Internal problem ID [25348]
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 : 10
Date solved : Friday, October 03, 2025 at 12:00:28 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 19

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

\[ y = c_1 \sin \left (2 \ln \left (t \right )\right )+c_2 \cos \left (2 \ln \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 22

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

\begin{align*} y(t)&\to c_1 \cos (2 \log (t))+c_2 \sin (2 \log (t)) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 19

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

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