67.3.20 problem Problem 21

Internal problem ID [13959]
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 : Wednesday, March 05, 2025 at 10:24:16 PM
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*}

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

ode:=diff(diff(diff(y(t),t),t),t)+8*diff(diff(y(t),t),t)+16*diff(y(t),t) = 0; 
ic:=y(0) = 1, D(y)(0) = 1, (D@@2)(y)(0) = -8; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 0.220 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(16*Derivative(y(t), t) + 8*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1, Subs(Derivative(y(t), (t, 2)), t, 0): -8} 
dsolve(ode,func=y(t),ics=ics)
 

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