68.13.32 problem 49

Internal problem ID [17685]
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 : 49
Date solved : Thursday, October 02, 2025 at 02:27:02 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (5\right )}+8 y^{\prime \prime \prime \prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=8 \\ y^{\prime }\left (0\right )&=4 \\ y^{\prime \prime }\left (0\right )&=0 \\ y^{\prime \prime \prime }\left (0\right )&=48 \\ y^{\prime \prime \prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 14

ode:=diff(diff(diff(diff(diff(y(t),t),t),t),t),t)+8*diff(diff(diff(diff(y(t),t),t),t),t) = 0; 
ic:=[y(0) = 8, D(y)(0) = 4, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 48, (D@@4)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = 8 t^{3}+4 t +8 \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 15

ode=D[ y[t],{t,5}]+8*D[y[t],{t,4}]==0; 
ic={y[0]==8,Derivative[1][y][0] ==4,Derivative[2][y][0] ==0,Derivative[3][y][0]==48,Derivative[4][y][0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 4 \left (2 t^3+t+2\right ) \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(8*Derivative(y(t), (t, 4)) + Derivative(y(t), (t, 5)),0) 
ics = {y(0): 8, Subs(Derivative(y(t), t), t, 0): 4, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 48, Subs(Derivative(y(t), (t, 4)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 8 t^{3} + 4 t + 8 \]