74.13.31 problem 48

Internal problem ID [16397]
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 : Tuesday, January 28, 2025 at 09:07:28 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*}

✓ Solution by Maple

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

dsolve([2*diff(y(t),t$5)+7*diff(y(t),t$4)+17*diff(y(t),t$3)+17*diff(y(t),t$2)+5*diff(y(t),t)=0,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],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{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,{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}},y[t],t,IncludeSingularSolutions -> True]
 

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