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 : Monday, January 27, 2025 at 06:12:14 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*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 24

dsolve([diff(y(t),t$4)-2*diff(y(t),t$3)+diff(y(t),t$2)+2*diff(y(t),t)-2*y(t)=0,exp(t)*cos(t)],singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[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,y[t],t,IncludeSingularSolutions -> True]
 

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