7.9.3 problem 3

Internal problem ID [2389]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 05:35:14 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 9 y^{\prime \prime }+6 y^{\prime }+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: 0.047 (sec). Leaf size: 14

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

\[ y = {\mathrm e}^{-\frac {t}{3}} \left (1+\frac {t}{3}\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 19

ode=9*D[y[t],{t,2}]+6*D[y[t],t]+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}{3} e^{-t/3} (t+3) \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(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): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {t}{3} + 1\right ) e^{- \frac {t}{3}} \]