7.7.7 problem 7

Internal problem ID [2370]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2, linear equations with constant coefficients. Page 138
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 05:34:58 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 5 y^{\prime \prime }+5 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: 0.112 (sec). Leaf size: 20

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

\[ y = \frac {2 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {5}\, \sinh \left (\frac {3 t \sqrt {5}}{10}\right )}{3} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 42

ode=5*D[y[t],{t,2}]+5*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 {1}{3} \sqrt {5} e^{-\frac {1}{10} \left (5+3 \sqrt {5}\right ) t} \left (e^{\frac {3 t}{\sqrt {5}}}-1\right ) \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 42

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

\[ y{\left (t \right )} = \frac {\sqrt {5} e^{\frac {t \left (-5 + 3 \sqrt {5}\right )}{10}}}{3} - \frac {\sqrt {5} e^{- \frac {t \left (5 + 3 \sqrt {5}\right )}{10}}}{3} \]