80.5.58 problem D 10

Internal problem ID [21279]
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 10
Date solved : Thursday, October 02, 2025 at 07:27:37 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{\prime \prime }-\frac {x^{\prime }}{t}&=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: 11

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

\[ x = \frac {t^{2}}{2}-\frac {1}{2} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 14

ode=D[x[t],{t,2}]-D[x[t],t]/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 {1}{2} \left (t^2-1\right ) \end{align*}

✓ Sympy. Time used: 0.078 (sec). Leaf size: 10

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

\[ x{\left (t \right )} = \frac {t^{2}}{2} - \frac {1}{2} \]