90.13.16 problem 16

Internal problem ID [25228]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 3. Second Order Constant Coefficient Linear Differential Equations. Exercises at page 235
Problem number : 16
Date solved : Thursday, October 02, 2025 at 11:59:08 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-5 \\ \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+13*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = -5]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 22

ode=D[y[t],{t,2}]+4*D[y[t],{t,1}]+13*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0] ==-5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} (\cos (3 t)-\sin (3 t)) \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(13*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -5} 
dsolve(ode,func=y(t),ics=ics)
 

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