68.13.15 problem 32

Internal problem ID [17668]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number : 32
Date solved : Thursday, October 02, 2025 at 02:26:57 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-2 y^{\prime \prime }+8 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+2*diff(diff(diff(y(t),t),t),t)-2*diff(diff(y(t),t),t)+8*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (c_3 \sin \left (t \right ) {\mathrm e}^{3 t}+c_4 \cos \left (t \right ) {\mathrm e}^{3 t}+c_2 t +c_1 \right ) {\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

ode=D[y[t],{t,4}]+2*D[ y[t],{t,3}]-2*D[y[t],{t,2}]+8*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} \left (c_4 t+c_2 e^{3 t} \cos (t)+c_1 e^{3 t} \sin (t)+c_3\right ) \end{align*}

✓ Sympy. Time used: 0.058 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(8*y(t) - 2*Derivative(y(t), (t, 2)) + 2*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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