80.5.65 problem D 16

Internal problem ID [21286]
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 5. Second order equations. Excercise 5.9 at page 119
Problem number : D 16
Date solved : Thursday, October 02, 2025 at 07:27:48 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 46

ode:=t^2*diff(diff(x(t),t),t)+a*t*diff(x(t),t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = t^{-\frac {a}{2}} \sqrt {t}\, \left (t^{\frac {\sqrt {a^{2}-2 a -3}}{2}} c_1 +t^{-\frac {\sqrt {a^{2}-2 a -3}}{2}} c_2 \right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 51

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

\begin{align*} x(t)&\to t^{\frac {1}{2} \left (-\sqrt {a^2-2 a-3}-a+1\right )} \left (c_2 t^{\sqrt {a^2-2 a-3}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.932 (sec). Leaf size: 474

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

\[ \text {Solution too large to show} \]