80.9.8 problem 8

Internal problem ID [21390]
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 : 8
Date solved : Thursday, October 02, 2025 at 07:30:42 PM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 32

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

\[ x = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1-\frac {1}{4} t^{2}+\frac {1}{64} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+\left (\frac {1}{4} t^{2}-\frac {3}{128} t^{4}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 83

ode=t^2*D[x[t],{t,2}]+t*D[x[t],t]+(t-1)*x[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},x[t],{t,0,5}]
 

\[ x(t)\to c_1 \left (\frac {31 t^4-176 t^3+144 t^2+576 t+576}{576 t}-\frac {1}{48} t \left (t^2-8 t+24\right ) \log (t)\right )+c_2 \left (\frac {t^5}{8640}-\frac {t^4}{360}+\frac {t^3}{24}-\frac {t^2}{3}+t\right ) \]

✓ Sympy. Time used: 0.267 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**2*Derivative(x(t), (t, 2)) + t*Derivative(x(t), t) + (t - 1)*x(t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ x{\left (t \right )} = C_{1} t \left (\frac {t^{4}}{8640} - \frac {t^{3}}{360} + \frac {t^{2}}{24} - \frac {t}{3} + 1\right ) + O\left (t^{6}\right ) \]