5.3.5 problem 12

Internal problem ID [1487]
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 : 12
Date solved : Tuesday, September 30, 2025 at 04:34:34 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 )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 0; 
ic:=[y(0) = 2, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-t} \left (4 \cos \left (2 t \right )+\sin \left (2 t \right )\right )}{2} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 25

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

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

✓ Sympy. Time used: 0.102 (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): -1} 
dsolve(ode,func=y(t),ics=ics)
 

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