52.6.11 problem 31

Internal problem ID [8346]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 31
Date solved : Wednesday, March 05, 2025 at 05:35:33 AM
CAS classification : [[_2nd_order, _missing_x]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.531 (sec). Leaf size: 18

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

\[ y = {\mathrm e}^{-t} \left (t -1+{\mathrm e} t +{\mathrm e}\right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 18

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

\[ y(t)\to e^{-t} (e t+t+e-1) \]

✓ Sympy. Time used: 0.170 (sec). Leaf size: 17

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

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