7.7.9 problem 9

Internal problem ID [2372]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2, linear equations with constant coefficients. Page 138
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 05:34:59 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.070 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+5*diff(y(t),t)+6*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = v]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 23

ode=D[y[t],{t,2}]+5*D[y[t],t]+6*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0] ==v}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-3 t} \left (e^t (v+3)-v-2\right ) \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(6*y(t) + 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): v} 
dsolve(ode,func=y(t),ics=ics)
 

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