70.13.33 problem 33

Internal problem ID [18902]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 33
Date solved : Thursday, October 02, 2025 at 03:32:39 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 11

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+9*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 2 \,{\mathrm e}^{-3 x} x \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 13

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

\begin{align*} y(x)&\to 2 e^{-3 x} x \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*y(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

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