67.19.6 problem 28.8 (c)

Internal problem ID [16891]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 28. The inverse Laplace transform. Additional Exercises. page 509
Problem number : 28.8 (c)
Date solved : Thursday, October 02, 2025 at 01:40:09 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+13 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=8 \\ \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+13*y(t) = 0; 
ic:=[y(0) = 2, D(y)(0) = 8]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-3 t} \left (2 \cos \left (2 t \right )+7 \sin \left (2 t \right )\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 24

ode=D[y[t],{t,2}]+6*D[y[t],t]+13*y[t]==0; 
ic={y[0]==2,Derivative[1][y][0] ==8}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-3 t} (7 \sin (2 t)+2 \cos (2 t)) \end{align*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(13*y(t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 8} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (7 \sin {\left (2 t \right )} + 2 \cos {\left (2 t \right )}\right ) e^{- 3 t} \]