89.12.22 problem 22

Internal problem ID [24576]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 121
Problem number : 22
Date solved : Thursday, October 02, 2025 at 10:46:12 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 14

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 0; 
ic:=[y(0) = 2, y(2) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 15

ode=D[y[x],{x,2}]+4*D[y[x],{x,1}]+ 4*y[x] ==0; 
ic={y[0]==2,y[2] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.090 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, y(2): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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