13.9.5 problem 6

Internal problem ID [2391]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:00:07 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1\\ y^{\prime }\left (2\right )&=-1 \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 10

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

\[ y = {\mathrm e}^{2-t} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 12

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

\[ y(t)\to e^{2-t} \]

✓ Sympy. Time used: 0.171 (sec). Leaf size: 8

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

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