67.11.22 problem 17.4 (d)

Internal problem ID [16623]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 17. Second order Homogeneous equations with constant coefficients. Additional exercises page 334
Problem number : 17.4 (d)
Date solved : Thursday, October 02, 2025 at 01:37:06 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 14

ode:=4*diff(diff(y(x),x),x)+4*diff(y(x),x)+y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x}{2}} \left (1+\frac {x}{2}\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 19

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

\begin{align*} y(x)&\to \frac {1}{2} e^{-x/2} (x+2) \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + 4*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {x}{2} + 1\right ) e^{- \frac {x}{2}} \]