80.9.4 problem 4

Internal problem ID [21386]
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 : 4
Date solved : Thursday, October 02, 2025 at 07:30:40 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

Order:=6; 
ode:=diff(diff(x(t),t),t)+t*diff(x(t),t)+x(t) = 0; 
ic:=[x(0) = 1, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t),type='series',t=0);
 

\[ x = 1-\frac {1}{2} t^{2}+\frac {1}{8} t^{4}+\operatorname {O}\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 19

ode=D[x[t],{t,2}]+t*D[x[t],t]+x[t]==0; 
ic={x[0]==1,Derivative[1][x][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},x[t],{t,0,5}]
 

\[ x(t)\to \frac {t^4}{8}-\frac {t^2}{2}+1 \]

✓ Sympy. Time used: 0.195 (sec). Leaf size: 29

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

\[ x{\left (t \right )} = C_{2} \left (\frac {t^{4}}{8} - \frac {t^{2}}{2} + 1\right ) + C_{1} t \left (1 - \frac {t^{2}}{3}\right ) + O\left (t^{6}\right ) \]