66.16.15 problem 15

Internal problem ID [16198]
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 : 15
Date solved : Thursday, October 02, 2025 at 10:43:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 20

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

\[ y = \frac {\left ({\mathrm e}^{t}+2\right ) \left ({\mathrm e}^{t}-1\right )^{2} {\mathrm e}^{-4 t}}{6} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 26

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

\begin{align*} y(t)&\to \frac {1}{6} e^{-4 t} \left (e^t-1\right )^2 \left (e^t+2\right ) \end{align*}

✓ Sympy. Time used: 0.179 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = \left (\frac {1}{6} - \frac {e^{- 2 t}}{2} + \frac {e^{- 3 t}}{3}\right ) e^{- t} \]