30.11.5 problem 5

Internal problem ID [7585]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 4, Linear Second-Order Equations. EXERCISES 4.1 at page 156
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 04:54:38 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+12 y&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+12*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 34

ode=1*D[y[t],{t,2}]+6*D[y[t],t]+12*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-3 t} \left (c_2 \cos \left (\sqrt {3} t\right )+c_1 \sin \left (\sqrt {3} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(12*y(t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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