68.13.33 problem 50

Internal problem ID [17686]
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 : 50
Date solved : Thursday, October 02, 2025 at 02:27:02 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (6\right )}-3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=16 \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime \prime }\left (0\right )&=0 \\ y^{\prime \prime \prime }\left (0\right )&=0 \\ y^{\prime \prime \prime \prime }\left (0\right )&=0 \\ y^{\left (5\right )}\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.079 (sec). Leaf size: 21

ode:=diff(diff(diff(diff(diff(diff(y(t),t),t),t),t),t),t)-3*diff(diff(diff(diff(y(t),t),t),t),t)+3*diff(diff(y(t),t),t)-y(t) = 0; 
ic:=[y(0) = 16, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0, (D@@4)(y)(0) = 0, (D@@5)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = 2 \cosh \left (t \right ) t^{2}-10 t \sinh \left (t \right )+16 \cosh \left (t \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 33

ode=D[ y[t],{t,6}]-3*D[y[t],{t,4}]+3*D[y[t],{t,2}]-y[t]==0; 
ic={y[0]==16,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0]==0,Derivative[4][y][0]==0,Derivative[5][y][0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} \left (t^2+e^{2 t} \left (t^2-5 t+8\right )+5 t+8\right ) \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) + 3*Derivative(y(t), (t, 2)) - 3*Derivative(y(t), (t, 4)) + Derivative(y(t), (t, 6)),0) 
ics = {y(0): 16, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 0, Subs(Derivative(y(t), (t, 4)), t, 0): 0, Subs(Derivative(y(t), (t, 5)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t \left (t - 5\right ) + 8\right ) e^{t} + \left (t \left (t + 5\right ) + 8\right ) e^{- t} \]