70.11.10 problem 24

Internal problem ID [18841]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.1 (Definitions and examples). Problems at page 214
Problem number : 24
Date solved : Thursday, October 02, 2025 at 03:31:36 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+y^{\prime }+16 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.108 (sec). Leaf size: 31

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+16*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{2}\right )+21 \cos \left (\frac {3 \sqrt {7}\, t}{2}\right )\right )}{21} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 47

ode=D[y[t],{t,2}]+1*D[y[t],t]+16*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{21} e^{-t/2} \left (\sqrt {7} \sin \left (\frac {3 \sqrt {7} t}{2}\right )+21 \cos \left (\frac {3 \sqrt {7} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(16*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {\sqrt {7} \sin {\left (\frac {3 \sqrt {7} t}{2} \right )}}{21} + \cos {\left (\frac {3 \sqrt {7} t}{2} \right )}\right ) e^{- \frac {t}{2}} \]