80.9.11 problem 11

Internal problem ID [21393]
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 : 11
Date solved : Thursday, October 02, 2025 at 07:30:44 PM
CAS classification : [_Bessel]

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

With initial conditions

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 25

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

\begin{align*} x(t)&\to c_1 \left (\operatorname {BesselJ}(m,t)-\frac {\operatorname {BesselJ}(m,0) \operatorname {BesselY}(m,t)}{\operatorname {BesselY}(m,0)}\right ) \end{align*}

✓ Sympy. Time used: 0.165 (sec). Leaf size: 39

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

\[ x{\left (t \right )} = C_{2} Y_{\sqrt {m^{2}}}\left (t\right ) - \frac {C_{2} J_{\sqrt {m^{2}}}\left (t\right ) Y_{\sqrt {m^{2}}}\left (0\right )}{J_{\sqrt {m^{2}}}\left (0\right )} \]