13.8.6 problem 5

Internal problem ID [2381]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.1, Complex roots. Page 141
Problem number : 5
Date solved : Tuesday, March 04, 2025 at 02:08:53 PM
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: 0.034 (sec). Leaf size: 32

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);
 

\[ y = \frac {{\mathrm e}^{-\frac {t}{2}} \left (5 \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right )+7 \cos \left (\frac {\sqrt {7}\, t}{2}\right )\right )}{7} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 48

ode=2*D[y[t],{t,2}]+3*D[y[t],t]+4*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{23} e^{-3 t/4} \left (11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{4}\right )+23 \cos \left (\frac {\sqrt {23} t}{4}\right )\right ) \]

✓ Sympy. Time used: 0.205 (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)
 

\[ y{\left (t \right )} = \left (\frac {5 \sqrt {7} \sin {\left (\frac {\sqrt {7} t}{2} \right )}}{7} + \cos {\left (\frac {\sqrt {7} t}{2} \right )}\right ) e^{- \frac {t}{2}} \]