80.9.9 problem 9

Internal problem ID [21391]
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 : 9
Date solved : Thursday, October 02, 2025 at 07:30:43 PM
CAS classification : [_Lienard]

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

With initial conditions

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

✗ Maple

ode:=t^2*diff(diff(x(t),t),t)+t*diff(x(t),t)+t^2*x(t) = 0; 
ic:=[D(x)(0) = a]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=t^2*D[x[t],{t,2}]+t*D[x[t],t]+t^2*x[t]==0; 
ic={Derivative[1][x][0] ==a}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

{}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 7

from sympy import * 
t = symbols("t") 
a = symbols("a") 
x = Function("x") 
ode = Eq(t**2*x(t) + t**2*Derivative(x(t), (t, 2)) + t*Derivative(x(t), t),0) 
ics = {Subs(Derivative(x(t), t), t, 0): a} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} J_{0}\left (t\right ) \]