80.9.2 problem 2

Internal problem ID [21384]
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 : 2
Date solved : Thursday, October 02, 2025 at 07:30:39 PM
CAS classification : [[_2nd_order, _missing_y]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 24

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

\[ x = c_1 \,t^{2} \left (1+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (-2+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 12

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

\[ x(t)\to c_2 t^2+c_1 \]

✓ Sympy. Time used: 0.165 (sec). Leaf size: 12

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

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