80.5.66 problem D 17

Internal problem ID [21287]
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 17
Date solved : Thursday, October 02, 2025 at 07:27:49 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} t^{2} x^{\prime \prime }-x^{\prime } t -3 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 14

ode:=t^2*diff(diff(x(t),t),t)-t*diff(x(t),t)-3*x(t) = 0; 
ic:=[x(1) = 0, D(x)(1) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {t^{4}-1}{4 t} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 17

ode=t^2*D[x[t],{t,2}]-t*D[x[t],t]-3*x[t]==0; 
ic={x[1]==0,Derivative[1][x][1] == 1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {t^4-1}{4 t} \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**2*Derivative(x(t), (t, 2)) - t*Derivative(x(t), t) - 3*x(t),0) 
ics = {x(1): 0, Subs(Derivative(x(t), t), t, 1): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {t^{3}}{4} - \frac {1}{4 t} \]