68.10.31 problem 31

Internal problem ID [17523]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.2, page 147
Problem number : 31
Date solved : Thursday, October 02, 2025 at 02:25:04 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.100 (sec). Leaf size: 20

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

\[ y = \frac {2 \,{\mathrm e}^{\frac {t}{2}} \sqrt {5}\, \sinh \left (\frac {t \sqrt {5}}{2}\right )}{5} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 38

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

\begin{align*} y(t)&\to \frac {e^{\frac {1}{2} \left (t-\sqrt {5} t\right )} \left (e^{\sqrt {5} t}-1\right )}{\sqrt {5}} \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 39

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

\[ y{\left (t \right )} = - \frac {\sqrt {5} e^{\frac {t \left (1 - \sqrt {5}\right )}{2}}}{5} + \frac {\sqrt {5} e^{\frac {t \left (1 + \sqrt {5}\right )}{2}}}{5} \]