13.8.2 problem 1

Internal problem ID [2377]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.1, Complex roots. Page 141
Problem number : 1
Date solved : Tuesday, March 04, 2025 at 02:08:42 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {t}{2}} \left (c_1 \sin \left (\frac {\sqrt {3}\, t}{2}\right )+c_2 \cos \left (\frac {\sqrt {3}\, t}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 42

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

\[ y(t)\to e^{-t/2} \left (c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]

✓ Sympy. Time used: 0.170 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} \]