80.9.10 problem 10

Internal problem ID [21392]
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 : 10
Date solved : Friday, October 03, 2025 at 07:51:55 AM
CAS classification : [_Bessel]

\begin{align*} t^{2} x^{\prime \prime }+x^{\prime } t +\left (t^{2}-1\right ) x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 10

ode:=t^2*diff(diff(x(t),t),t)+t*diff(x(t),t)+(t^2-1)*x(t) = 0; 
ic:=[D(x)(0) = a]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = 2 a \operatorname {BesselJ}\left (1, t\right ) \]

✓ Mathematica. Time used: 0.47 (sec). Leaf size: 11

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

\begin{align*} x(t)&\to 2 a \operatorname {BesselJ}(1,t) \end{align*}

✗ Sympy

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

ValueError : Couldnt solve for initial conditions