72.16.4 problem 4

Internal problem ID [14821]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.1 page 399
Problem number : 4
Date solved : Thursday, March 13, 2025 at 04:20:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

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

\[ y = c_{2} {\mathrm e}^{-2 t} \sin \left (3 t \right )+c_{1} {\mathrm e}^{-2 t} \cos \left (3 t \right )+\frac {{\mathrm e}^{-t}}{10} \]

✓ Mathematica. Time used: 0.106 (sec). Leaf size: 76

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

\[ y(t)\to e^{-2 t} \left (\cos (3 t) \int _1^t-\frac {1}{3} e^{K[2]} \sin (3 K[2])dK[2]+\sin (3 t) \int _1^t\frac {1}{3} e^{K[1]} \cos (3 K[1])dK[1]+c_2 \cos (3 t)+c_1 \sin (3 t)\right ) \]

✓ Sympy. Time used: 0.249 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = \left (\left (C_{1} \sin {\left (3 t \right )} + C_{2} \cos {\left (3 t \right )}\right ) e^{- t} + \frac {1}{10}\right ) e^{- t} \]