80.9.13 problem 21

Internal problem ID [21395]
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 : 21
Date solved : Thursday, October 02, 2025 at 07:30:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 100

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

\begin{align*} x(t)&\to \frac {2 \operatorname {BesselJ}\left (\sqrt {\lambda },0\right ) \operatorname {BesselY}\left (\sqrt {\lambda },t\right )-2 \operatorname {BesselY}\left (\sqrt {\lambda },0\right ) \operatorname {BesselJ}\left (\sqrt {\lambda },t\right )}{\operatorname {BesselJ}\left (\sqrt {\lambda },0\right ) \left (\operatorname {BesselY}\left (\sqrt {\lambda }-1,0\right )-\operatorname {BesselY}\left (\sqrt {\lambda }+1,0\right )\right )+\left (\operatorname {BesselJ}\left (\sqrt {\lambda }+1,0\right )-\operatorname {BesselJ}\left (\sqrt {\lambda }-1,0\right )\right ) \operatorname {BesselY}\left (\sqrt {\lambda },0\right )} \end{align*}

✓ Sympy. Time used: 0.181 (sec). Leaf size: 155

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

\[ x{\left (t \right )} = \frac {2 J_{\sqrt {\lambda _{}}}\left (0\right ) Y_{\sqrt {\lambda _{}}}\left (t\right )}{J_{\sqrt {\lambda _{}}}\left (0\right ) Y_{\sqrt {\lambda _{}} - 1}\left (0\right ) - J_{\sqrt {\lambda _{}}}\left (0\right ) Y_{\sqrt {\lambda _{}} + 1}\left (0\right ) - J_{\sqrt {\lambda _{}} - 1}\left (0\right ) Y_{\sqrt {\lambda _{}}}\left (0\right ) + J_{\sqrt {\lambda _{}} + 1}\left (0\right ) Y_{\sqrt {\lambda _{}}}\left (0\right )} - \frac {2 J_{\sqrt {\lambda _{}}}\left (t\right ) Y_{\sqrt {\lambda _{}}}\left (0\right )}{J_{\sqrt {\lambda _{}}}\left (0\right ) Y_{\sqrt {\lambda _{}} - 1}\left (0\right ) - J_{\sqrt {\lambda _{}}}\left (0\right ) Y_{\sqrt {\lambda _{}} + 1}\left (0\right ) - J_{\sqrt {\lambda _{}} - 1}\left (0\right ) Y_{\sqrt {\lambda _{}}}\left (0\right ) + J_{\sqrt {\lambda _{}} + 1}\left (0\right ) Y_{\sqrt {\lambda _{}}}\left (0\right )} \]