80.9.5 problem 5

Internal problem ID [21387]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 9. Solutions by infinite series and Bessel functions. Excercise 10.6 at page 223
Problem number : 5
Date solved : Thursday, October 02, 2025 at 07:30:40 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

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

Order:=6; 
ode:=4*t^2*diff(diff(x(t),t),t)+4*t*diff(x(t),t)-x(t) = 0; 
dsolve(ode,x(t),type='series',t=0);
 

\[ x = \frac {t c_1 +c_2}{\sqrt {t}}+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 20

ode=4*t^2*D[x[t],{t,2}]+4*t*D[x[t],t]-x[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},x[t],{t,0,5}]
 

\[ x(t)\to \frac {c_1}{\sqrt {t}}+c_2 \sqrt {t} \]

✓ Sympy. Time used: 0.215 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*t**2*Derivative(x(t), (t, 2)) + 4*t*Derivative(x(t), t) - x(t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ x{\left (t \right )} = C_{2} \sqrt {t} + \frac {C_{1}}{\sqrt {t}} + O\left (t^{6}\right ) \]