14.21.28 problem Problem 28

Internal problem ID [3955]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.4. page 689
Problem number : Problem 28
Date solved : Tuesday, September 30, 2025 at 06:59:25 AM
CAS classification : [[_2nd_order, _missing_x]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.106 (sec). Leaf size: 13

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

\[ y = A \cosh \left (t \right )+B \sinh \left (t \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 0.040 (sec). Leaf size: 22

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

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