5.3.1 problem 8

Internal problem ID [1483]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.2, The Laplace Transform. Solution of Initial Value Problems. page 255
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 04:34:32 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-y^{\prime }-6 y&=0 \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: 0.103 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)-diff(y(t),t)-6*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{3 t}}{5}+\frac {4 \,{\mathrm e}^{-2 t}}{5} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 21

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

\begin{align*} y(t)&\to \frac {1}{5} e^{-2 t} \left (e^{5 t}+4\right ) \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*y(t) - Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {e^{3 t}}{5} + \frac {4 e^{- 2 t}}{5} \]