74.13.30 problem 47

Internal problem ID [16317]
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 : 47
Date solved : Thursday, March 13, 2025 at 08:10:21 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} 8 y^{\left (5\right )}+4 y^{\prime \prime \prime \prime }+66 y^{\prime \prime \prime }-41 y^{\prime \prime }-37 y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=-14\\ y^{\prime \prime }\left (0\right )&=-14\\ y^{\prime \prime \prime }\left (0\right )&=139\\ y^{\prime \prime \prime \prime }\left (0\right )&=-{\frac {29}{4}} \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 23

ode:=8*diff(diff(diff(diff(diff(y(t),t),t),t),t),t)+4*diff(diff(diff(diff(y(t),t),t),t),t)+66*diff(diff(diff(y(t),t),t),t)-41*diff(diff(y(t),t),t)-37*diff(y(t),t) = 0; 
ic:=y(0) = 4, D(y)(0) = -14, (D@@2)(y)(0) = -14, (D@@3)(y)(0) = 139, (D@@4)(y)(0) = -29/4; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {t}{2}} \left (1-4 \sin \left (3 t \right )+3 \cos \left (3 t \right )\right ) \]

✓ Mathematica. Time used: 0.241 (sec). Leaf size: 79

ode=8*D[ y[t],{t,5}]+4*D[y[t],{t,4}]+66*D[ y[t],{t,3}]-41*D[y[t],{t,2}]-37*D[y[t],t]==0; 
ic={y[0]==4,Derivative[1][y][0] ==-14,Derivative[2][y][0] ==-14,Derivative[3][y][0]==139,Derivative[4][y][0]==-29/4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \int _1^t-\frac {1}{2} e^{-\frac {K[1]}{2}} (27 \cos (3 K[1])+14 \sin (3 K[1])+1)dK[1]-\int _1^0-\frac {1}{2} e^{-\frac {K[1]}{2}} (27 \cos (3 K[1])+14 \sin (3 K[1])+1)dK[1]+4 \]

✓ Sympy. Time used: 0.353 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-37*Derivative(y(t), t) - 41*Derivative(y(t), (t, 2)) + 66*Derivative(y(t), (t, 3)) + 4*Derivative(y(t), (t, 4)) + 8*Derivative(y(t), (t, 5)),0) 
ics = {y(0): 4, Subs(Derivative(y(t), t), t, 0): -14, Subs(Derivative(y(t), (t, 2)), t, 0): -14, Subs(Derivative(y(t), (t, 3)), t, 0): 139, Subs(Derivative(y(t), (t, 4)), t, 0): -29/4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- 4 \sin {\left (3 t \right )} + 3 \cos {\left (3 t \right )} + 1\right ) e^{- \frac {t}{2}} \]