58.20.6 problem 6

Internal problem ID [14930]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:56:39 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 0; 
ic:=[y(0) = 2, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-t} \left (2 \cos \left (2 t \right )+3 \sin \left (2 t \right )\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 24

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==0; 
ic={y[0]==2,Derivative[1][y][0]==4}; 
DSolve[{ode,ic},{y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} (3 \sin (2 t)+2 \cos (2 t)) \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (3 \sin {\left (2 t \right )} + 2 \cos {\left (2 t \right )}\right ) e^{- t} \]