14.21.6 problem 6

Internal problem ID [2715]
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 : 6
Date solved : Monday, January 27, 2025 at 06:12:14 AM
CAS classification : [[_high_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 19

dsolve([diff(y(t),t$4)-y(t)=0,y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = -1],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-t}}{2}+\frac {\sin \left (t \right )}{2}+\frac {\cos \left (t \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 19

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

\[ y(t)\to \frac {1}{2} \left (e^{-t}+\sin (t)+\cos (t)\right ) \]