14.7.5 problem 5

Internal problem ID [2549]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2. Linear equations with constant coefficients. Excercises page 140
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:09:52 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }-4 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: 0.050 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t)-4*y(t) = 0; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{4 t}}{5}+\frac {4 \,{\mathrm e}^{-t}}{5} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 21

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

\[ y(t)\to \frac {1}{5} e^{-t} \left (e^{5 t}+4\right ) \]

✓ Sympy. Time used: 0.173 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*y(t) - 3*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)
 

\[ y{\left (t \right )} = \frac {e^{4 t}}{5} + \frac {4 e^{- t}}{5} \]