74.13.26 problem 43

Internal problem ID [16392]
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 : 43
Date solved : Tuesday, January 28, 2025 at 09:07:26 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+16 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=8\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 17

dsolve([diff(y(t),t$4)-8*diff(y(t),t$2)+16*y(t)=0,y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 8, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{D[y[t],{t,4}]-8*D[y[t],{t,2}]+16*y[t]==0,{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==8,Derivative[3][y][0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-2 t} \left (e^{4 t}-1\right ) t \]