67.4.3 problem Problem 2(c)

Internal problem ID [13976]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(c)
Date solved : Monday, March 31, 2025 at 08:20:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{\prime \prime }+5 y^{\prime }+4 y&=3 \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.132 (sec). Leaf size: 38

ode:=4*diff(diff(y(t),t),t)+5*diff(y(t),t)+4*y(t) = 3*exp(-t); 
ic:=y(0) = -1, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {2 \sin \left (\frac {\sqrt {39}\, t}{8}\right ) {\mathrm e}^{-\frac {5 t}{8}} \sqrt {39}}{13}-2 \,{\mathrm e}^{-\frac {5 t}{8}} \cos \left (\frac {\sqrt {39}\, t}{8}\right )+{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 58

ode=4*D[y[t],{t,2}]+5*D[y[t],t]+4*y[t]==3*Exp[-t]; 
ic={y[0]==-1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-t}+2 \sqrt {\frac {3}{13}} e^{-5 t/8} \sin \left (\frac {\sqrt {39} t}{8}\right )-2 e^{-5 t/8} \cos \left (\frac {\sqrt {39} t}{8}\right ) \]

✓ Sympy. Time used: 0.286 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + 5*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)) - 3*exp(-t),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {2 \sqrt {39} \sin {\left (\frac {\sqrt {39} t}{8} \right )}}{13} - 2 \cos {\left (\frac {\sqrt {39} t}{8} \right )}\right ) e^{- \frac {5 t}{8}} + e^{- t} \]