72.16.11 problem 11

Internal problem ID [14828]
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 : 11
Date solved : Thursday, March 13, 2025 at 04:20:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=-3 \,{\mathrm e}^{-2 t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+13*y(t) = -3*exp(-2*t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 20

ode=D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==-3*Exp[-2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{3} e^{-2 t} (\cos (3 t)-1) \]

✓ Sympy. Time used: 0.283 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(13*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 3*exp(-2*t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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