2.7.9 problem 9

Internal problem ID [815]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.1, second order linear equations. Page 299
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 04:15:41 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 (0\right )&=2 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 14

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

\begin{align*} y(x)&\to e^{-x} (x+2) \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 8

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

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