52.5.5 problem 35

Internal problem ID [8328]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.2.2 TRANSFORMS OF DERIVATIVES Page 289
Problem number : 35
Date solved : Wednesday, March 05, 2025 at 05:35:16 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+4 y&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.651 (sec). Leaf size: 17

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

\[ y = \frac {4 \,{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{-4 t}}{3} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 23

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

\[ y(t)\to \frac {1}{3} e^{-4 t} \left (4 e^{3 t}-1\right ) \]

✓ Sympy. Time used: 0.212 (sec). Leaf size: 17

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

\[ y{\left (t \right )} = \left (\frac {4}{3} - \frac {e^{- 3 t}}{3}\right ) e^{- t} \]