10.9.13 problem 13

Internal problem ID [1315]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.4 Repeated roots, reduction of order , page 172
Problem number : 13
Date solved : Saturday, March 29, 2025 at 10:52:01 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 9 y^{\prime \prime }+6 y^{\prime }+82 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: 0.073 (sec). Leaf size: 22

ode:=9*diff(diff(y(t),t),t)+6*diff(y(t),t)+82*y(t) = 0; 
ic:=y(0) = -1, D(y)(0) = 2; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {t}{3}} \left (\frac {5 \sin \left (3 t \right )}{9}-\cos \left (3 t \right )\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 29

ode=9*D[y[t],{t,2}]+6*D[y[t],t]+82*y[t]==0; 
ic={y[0]==-1,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{9} e^{-t/3} (5 \sin (3 t)-9 \cos (3 t)) \]

✓ Sympy. Time used: 0.183 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(82*y(t) + 6*Derivative(y(t), t) + 9*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)
 

\[ y{\left (t \right )} = \left (\frac {5 \sin {\left (3 t \right )}}{9} - \cos {\left (3 t \right )}\right ) e^{- \frac {t}{3}} \]