74.10.28 problem 28

Internal problem ID [16162]
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 : 28
Date solved : Monday, March 31, 2025 at 02:45:39 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 )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.111 (sec). Leaf size: 34

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

\[ y = \frac {{\mathrm e}^{-\frac {\left (\sqrt {5}+1\right ) t}{2}} \left (\left (5+\sqrt {5}\right ) {\mathrm e}^{t \sqrt {5}}+5-\sqrt {5}\right )}{10} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 49

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

\[ y(t)\to \frac {1}{10} e^{-\frac {1}{2} \left (1+\sqrt {5}\right ) t} \left (\left (5+\sqrt {5}\right ) e^{\sqrt {5} t}+5-\sqrt {5}\right ) \]

✓ Sympy. Time used: 0.188 (sec). Leaf size: 46

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): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {\sqrt {5}}{10} + \frac {1}{2}\right ) e^{\frac {t \left (-1 + \sqrt {5}\right )}{2}} + \left (\frac {1}{2} - \frac {\sqrt {5}}{10}\right ) e^{- \frac {t \left (1 + \sqrt {5}\right )}{2}} \]