76.24.11 problem 13

Internal problem ID [17739]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.2 (Basic Theory of First Order Linear Systems). Problems at page 398
Problem number : 13
Date solved : Monday, March 31, 2025 at 04:26:18 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+4 y^{\prime \prime }-4 y^{\prime }-16 y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

ode:=diff(diff(diff(y(t),t),t),t)+4*diff(diff(y(t),t),t)-4*diff(y(t),t)-16*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (c_1 \,{\mathrm e}^{6 t}+c_3 \,{\mathrm e}^{2 t}+c_2 \right ) {\mathrm e}^{-4 t} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 30

ode=D[y[t],{t,3}]+4*D[y[t],{t,2}]-4*D[y[t],t]-16*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-4 t} \left (c_2 e^{2 t}+c_3 e^{6 t}+c_1\right ) \]

✓ Sympy. Time used: 0.181 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-16*y(t) - 4*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{- 2 t} + C_{3} e^{2 t} \]