80.9.1 problem 1

Internal problem ID [21383]
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 : 1
Date solved : Thursday, October 02, 2025 at 07:30:38 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 58

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

\[ x = c_1 t \left (1+\frac {1}{2} t +\frac {1}{12} t^{2}+\frac {1}{144} t^{3}+\frac {1}{2880} t^{4}+\frac {1}{86400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\ln \left (t \right ) \left (t +\frac {1}{2} t^{2}+\frac {1}{12} t^{3}+\frac {1}{144} t^{4}+\frac {1}{2880} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (1-\frac {3}{4} t^{2}-\frac {7}{36} t^{3}-\frac {35}{1728} t^{4}-\frac {101}{86400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 85

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

\[ x(t)\to c_1 \left (\frac {1}{144} t \left (t^3+12 t^2+72 t+144\right ) \log (t)+\frac {-47 t^4-480 t^3-2160 t^2-1728 t+1728}{1728}\right )+c_2 \left (\frac {t^5}{2880}+\frac {t^4}{144}+\frac {t^3}{12}+\frac {t^2}{2}+t\right ) \]

✓ Sympy. Time used: 0.179 (sec). Leaf size: 29

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

\[ x{\left (t \right )} = C_{1} t \left (\frac {t^{4}}{2880} + \frac {t^{3}}{144} + \frac {t^{2}}{12} + \frac {t}{2} + 1\right ) + O\left (t^{6}\right ) \]