76.19.1 problem 1

Internal problem ID [17646]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 1
Date solved : Thursday, March 13, 2025 at 10:45:54 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }-12 y&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 10.803 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)-12*y(t) = 0; 
ic:=y(0) = 8, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = 2 \,{\mathrm e}^{6 t}+6 \,{\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 19

ode=D[y[t],{t,2}]-4*D[y[t],t]-12*y[t]==0; 
ic={y[0]==8,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to 2 e^{-2 t} \left (e^{8 t}+3\right ) \]

✓ Sympy. Time used: 0.163 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-12*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 8, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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