44.9.22 problem 2(d)

Internal problem ID [9244]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.1. Linear Equations with Constant Coefficients. Page 62
Problem number : 2(d)
Date solved : Tuesday, September 30, 2025 at 06:15:35 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 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: 0.064 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+5*y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 18

ode=D[y[x],{x,2}]+4*D[y[x],x]+5*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 e^{-2 x} (2 \sin (x)+\cos (x)) \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) + 4*Derivative(y(x), x) + 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)
 

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