80.9.3 problem 3

Internal problem ID [21385]
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 : 3
Date solved : Thursday, October 02, 2025 at 07:30:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} t x^{\prime \prime }&=x t +1 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

ValueError : ODE -t*x(t) + t*Derivative(x(t), (t, 2)) - 1 does not match hint 2nd_power_series_regular