14.8.5 problem 5

Internal problem ID [2560]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.1 Linear equations with constant coefficients (complex roots). Excercises page 144
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:10:11 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.102 (sec). Leaf size: 29

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

\[ y = {\mathrm e}^{-\frac {t}{2}} \left (-\frac {3 \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right )}{7}+\cos \left (\frac {\sqrt {7}\, t}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 48

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

\[ y(t)\to \frac {1}{7} e^{-t/2} \left (7 \cos \left (\frac {\sqrt {7} t}{2}\right )-3 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \]

✓ Sympy. Time used: 0.177 (sec). Leaf size: 36

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

\[ y{\left (t \right )} = \left (- \frac {3 \sqrt {7} \sin {\left (\frac {\sqrt {7} t}{2} \right )}}{7} + \cos {\left (\frac {\sqrt {7} t}{2} \right )}\right ) e^{- \frac {t}{2}} \]