76.18.1 problem 12

Internal problem ID [17641]
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 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 12
Date solved : Monday, March 31, 2025 at 04:23:25 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }-2 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.118 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)-2*y(t) = 0; 
ic:=y(0) = 2, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-t} \left (\sinh \left (t \sqrt {3}\right ) \sqrt {3}+2 \cosh \left (t \sqrt {3}\right )\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 48

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

\[ y(t)\to \frac {1}{2} e^{-\left (\left (1+\sqrt {3}\right ) t\right )} \left (\left (2+\sqrt {3}\right ) e^{2 \sqrt {3} t}+2-\sqrt {3}\right ) \]

✓ Sympy. Time used: 0.194 (sec). Leaf size: 39

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

\[ y{\left (t \right )} = \left (\frac {\sqrt {3}}{2} + 1\right ) e^{t \left (-1 + \sqrt {3}\right )} + \left (1 - \frac {\sqrt {3}}{2}\right ) e^{- t \left (1 + \sqrt {3}\right )} \]