80.5.67 problem D 18

Internal problem ID [21288]
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 18
Date solved : Thursday, October 02, 2025 at 07:27:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 18

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

\[ x = \sin \left (\ln \left (t \right )\right ) c_2 +\cos \left (\ln \left (t \right )\right ) c_1 +\frac {t}{2} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 23

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

\begin{align*} x(t)&\to \frac {t}{2}+c_1 \cos (\log (t))+c_2 \sin (\log (t)) \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 19

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

\[ x{\left (t \right )} = C_{1} \sin {\left (\log {\left (t \right )} \right )} + C_{2} \cos {\left (\log {\left (t \right )} \right )} + \frac {t}{2} \]