90.22.13 problem 13

Internal problem ID [25351]
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 : 13
Date solved : Friday, October 03, 2025 at 12:00:31 AM
CAS classification : [[_Emden, _Fowler]]

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

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 14

ode:=4*t^2*diff(diff(y(t),t),t)+y(t) = 0; 
ic:=[y(1) = 2, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \left (-\ln \left (t \right )+2\right ) \sqrt {t} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 16

ode=4*t^2*D[y[t],{t,2}]+y[t]==0; 
ic={y[1]==2,Derivative[1][y][1] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sqrt {t} (\log (t)-2) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*t**2*Derivative(y(t), (t, 2)) + y(t),0) 
ics = {y(1): 2, Subs(Derivative(y(t), t), t, 1): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \sqrt {t} \left (2 - \log {\left (t \right )}\right ) \]