68.13.37 problem 63 (b)

Internal problem ID [17690]
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 : 63 (b)
Date solved : Thursday, October 02, 2025 at 02:27:04 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+30 y^{\prime \prime }-56 y^{\prime }+49 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=2 \\ y^{\prime \prime }\left (0\right )&=-1 \\ y^{\prime \prime \prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 36

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-8*diff(diff(diff(y(t),t),t),t)+30*diff(diff(y(t),t),t)-56*diff(y(t),t)+49*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = 2, (D@@2)(y)(0) = -1, (D@@3)(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {\left (\left (\frac {21 t}{2}-3\right ) \cos \left (\sqrt {3}\, t \right )+\sqrt {3}\, \sin \left (\sqrt {3}\, t \right ) \left (t -\frac {7}{2}\right )\right ) {\mathrm e}^{2 t}}{3} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 49

ode=D[y[t],{t,4}]-8*D[ y[t],{t,3}]+30*D[y[t],{t,2}]-56*D[y[t],t]+49*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0] ==2,Derivative[2][y][0] ==-1,Derivative[3][y][0]==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\frac {1}{6} e^{2 t} \left (\sqrt {3} (2 t-7) \sin \left (\sqrt {3} t\right )+3 (7 t-2) \cos \left (\sqrt {3} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.230 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(49*y(t) - 56*Derivative(y(t), t) + 30*Derivative(y(t), (t, 2)) - 8*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 2, Subs(Derivative(y(t), (t, 2)), t, 0): -1, Subs(Derivative(y(t), (t, 3)), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (1 - \frac {7 t}{2}\right ) \cos {\left (\sqrt {3} t \right )} + \left (- \frac {\sqrt {3} t}{3} + \frac {7 \sqrt {3}}{6}\right ) \sin {\left (\sqrt {3} t \right )}\right ) e^{2 t} \]