7.8.1 problem Example 2

Internal problem ID [2376]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.1, Complex roots. Page 141
Problem number : Example 2
Date solved : Tuesday, September 30, 2025 at 05:35:04 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+4 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.102 (sec). Leaf size: 30

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+4*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 40

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

\begin{align*} y(t)&\to \frac {1}{3} e^{-t} \left (2 \sqrt {3} \sin \left (\sqrt {3} t\right )+3 \cos \left (\sqrt {3} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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