81.10.14 problem 14-14

Internal problem ID [21609]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 14. Second order homogeneous differential equations with constant coefficients. Page 297.
Problem number : 14-14
Date solved : Thursday, October 02, 2025 at 07:58:59 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 37

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

\[ y = \frac {\left (\left (3-2 \sqrt {3}\right ) {\mathrm e}^{2 x \sqrt {3}}+2 \sqrt {3}+3\right ) {\mathrm e}^{-\left (\sqrt {3}-1\right ) x}}{6} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to \frac {1}{6} e^{x-\sqrt {3} x} \left (\left (3-2 \sqrt {3}\right ) e^{2 \sqrt {3} x}+3+2 \sqrt {3}\right ) \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = \left (\frac {1}{2} + \frac {\sqrt {3}}{3}\right ) e^{x \left (1 - \sqrt {3}\right )} + \left (\frac {1}{2} - \frac {\sqrt {3}}{3}\right ) e^{x \left (1 + \sqrt {3}\right )} \]