14.21.8 problem 8

Internal problem ID [2717]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.15, Higher order equations. Excercises page 263
Problem number : 8
Date solved : Sunday, March 30, 2025 at 12:15:35 AM
CAS classification : [[_high_order, _missing_x]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{t} \cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 36

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

\[ y(t)\to c_3 e^{-t}+c_4 e^t+c_2 e^t \cos (t)+c_1 e^t \sin (t) \]

✓ Sympy. Time used: 0.208 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*y(t) + 2*Derivative(y(t), t) + 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)
 

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