74.13.31 problem 48

Internal problem ID [16318]
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 : 48
Date solved : Thursday, March 13, 2025 at 08:10:22 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} 2 y^{\left (5\right )}+7 y^{\prime \prime \prime \prime }+17 y^{\prime \prime \prime }+17 y^{\prime \prime }+5 y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-3\\ y^{\prime }\left (0\right )&={\frac {15}{2}}\\ y^{\prime \prime }\left (0\right )&={\frac {17}{4}}\\ y^{\prime \prime \prime }\left (0\right )&=-{\frac {385}{8}}\\ y^{\prime \prime \prime \prime }\left (0\right )&={\frac {1217}{16}} \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 27

ode:=2*diff(diff(diff(diff(diff(y(t),t),t),t),t),t)+7*diff(diff(diff(diff(y(t),t),t),t),t)+17*diff(diff(diff(y(t),t),t),t)+17*diff(diff(y(t),t),t)+5*diff(y(t),t) = 0; 
ic:=y(0) = -3, D(y)(0) = 15/2, (D@@2)(y)(0) = 17/4, (D@@3)(y)(0) = -385/8, (D@@4)(y)(0) = 1217/16; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.222 (sec). Leaf size: 89

ode=2*D[ y[t],{t,5}]+7*D[y[t],{t,4}]+17*D[ y[t],{t,3}]+17*D[y[t],{t,2}]+5*D[y[t],t]==0; 
ic={y[0]==-3,Derivative[1][y][0] ==15/2,Derivative[2][y][0] ==17/4,Derivative[3][y][0]==-385/8,Derivative[4][y][0]==1217/16}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \int _1^t-\frac {1}{2} e^{-K[1]} \left (-16 \cos (2 K[1])+e^{\frac {K[1]}{2}}-12 \sin (2 K[1])\right )dK[1]-\int _1^0-\frac {1}{2} e^{-K[1]} \left (-16 \cos (2 K[1])+e^{\frac {K[1]}{2}}-12 \sin (2 K[1])\right )dK[1]-3 \]

✓ Sympy. Time used: 0.362 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*Derivative(y(t), t) + 17*Derivative(y(t), (t, 2)) + 17*Derivative(y(t), (t, 3)) + 7*Derivative(y(t), (t, 4)) + 2*Derivative(y(t), (t, 5)),0) 
ics = {y(0): -3, Subs(Derivative(y(t), t), t, 0): 15/2, Subs(Derivative(y(t), (t, 2)), t, 0): 17/4, Subs(Derivative(y(t), (t, 3)), t, 0): -385/8, Subs(Derivative(y(t), (t, 4)), t, 0): 1217/16} 
dsolve(ode,func=y(t),ics=ics)
 

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