80.5.33 problem C 9

Internal problem ID [21254]
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 : C 9
Date solved : Thursday, October 02, 2025 at 07:27:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }-4 x^{\prime }+13 x&=20 \,{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

ode:=diff(diff(x(t),t),t)-4*diff(x(t),t)+13*x(t) = 20*exp(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{2 t} \sin \left (3 t \right ) c_2 +{\mathrm e}^{2 t} \cos \left (3 t \right ) c_1 +2 \,{\mathrm e}^{t} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 31

ode=D[x[t],{t,2}]-4*D[x[t],t]+13*x[t]==20*Exp[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^t \left (c_2 e^t \cos (3 t)+c_1 e^t \sin (3 t)+2\right ) \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(13*x(t) - 20*exp(t) - 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\left (C_{1} \sin {\left (3 t \right )} + C_{2} \cos {\left (3 t \right )}\right ) e^{t} + 2\right ) e^{t} \]