80.5.57 problem D 7

Internal problem ID [21278]
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 7
Date solved : Thursday, October 02, 2025 at 07:27:36 PM
CAS classification : [_Lienard]

\begin{align*} x^{\prime \prime }-\frac {x^{\prime } t}{4}+x&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 48

ode:=diff(diff(x(t),t),t)-1/4*t*diff(x(t),t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = -4 t c_1 \left (t^{2}-20\right ) {\mathrm e}^{\frac {t^{2}}{8}}+\left (t^{4}-24 t^{2}+48\right ) \left (\operatorname {erfi}\left (\frac {\sqrt {2}\, t}{4}\right ) \sqrt {2}\, \sqrt {\pi }\, c_1 +c_2 \right ) \]

✓ Mathematica. Time used: 0.98 (sec). Leaf size: 72

ode=D[x[t],{t,2}]-1/4*t*D[x[t],t]+x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {c_2 \left (\sqrt {2 \pi } \left (t^4-24 t^2+48\right ) \text {erfi}\left (\frac {t}{2 \sqrt {2}}\right )-4 e^{\frac {t^2}{8}} t \left (t^2-20\right )\right )}{6144}+c_1 \left (t^4-24 t^2+48\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t*Derivative(x(t), t)/4 + x(t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

False