67.3.20 problem Problem 21

Internal problem ID [14038]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number : Problem 21
Date solved : Tuesday, January 28, 2025 at 06:13:06 AM
CAS classification : [[_3rd_order, _missing_x]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1\\ y^{\prime \prime }\left (0\right )&=-8 \end{align*}

✓ Solution by Maple

Time used: 8.708 (sec). Leaf size: 12

dsolve([diff(y(t),t$3)+8*diff(y(t),t$2)+16*diff(y(t),t)=0,y(0) = 1, D(y)(0) = 1, (D@@2)(y)(0) = -8],y(t), singsol=all)
 

\[ y = t \,{\mathrm e}^{-4 t}+1 \]

✓ Solution by Mathematica

Time used: 4.161 (sec). Leaf size: 49

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

\[ y(t)\to \int _1^te^{-4 K[1]} (1-4 K[1])dK[1]-\int _1^0e^{-4 K[1]} (1-4 K[1])dK[1]+1 \]