72.12.23 problem 2 (e)

Internal problem ID [19596]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 17. The Homogeneous Equation with Constant Coefficients. Problems at page 125
Problem number : 2 (e)
Date solved : Thursday, October 02, 2025 at 04:40:27 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+2 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=2+3 \sqrt {2} \\ \end{align*}

✓ Maple. Time used: 0.095 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+2*y(x) = 0; 
ic:=[y(0) = -1, D(y)(0) = 2+3*2^(1/2)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 30

ode=D[y[x],{x,2}] +4*D[y[x],x]+2*y[x]==0; 
ic={y[0]==-1,Derivative[1][y][0] == 2+3*Sqrt[2]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.121 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 2 + 3*sqrt(2)} 
dsolve(ode,func=y(x),ics=ics)
 

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