68.13.25 problem 42

Internal problem ID [17678]
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 : 42
Date solved : Thursday, October 02, 2025 at 02:27:00 PM
CAS classification : [[_high_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 20

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+16*diff(diff(diff(y(t),t),t),t) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {1}{4096}+\frac {255 t}{256}+\frac {t^{2}}{32}-\frac {{\mathrm e}^{-16 t}}{4096} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 26

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

\begin{align*} y(t)&\to \frac {128 t^2+4080 t-e^{-16 t}+1}{4096} \end{align*}

✓ Sympy. Time used: 0.078 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(16*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t^{2}}{32} + \frac {255 t}{256} + \frac {1}{4096} - \frac {e^{- 16 t}}{4096} \]