68.13.28 problem 45

Internal problem ID [17681]
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 : 45
Date solved : Thursday, October 02, 2025 at 02:27:01 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=3 \\ y^{\prime \prime }\left (0\right )&=-7 \\ y^{\prime \prime \prime }\left (0\right )&=15 \\ \end{align*}

✓ Maple. Time used: 0.060 (sec). Leaf size: 15

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-5*diff(diff(y(t),t),t)+4*y(t) = 0; 
ic:=[y(0) = -1, D(y)(0) = 3, (D@@2)(y)(0) = -7, (D@@3)(y)(0) = 15]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 16

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

\begin{align*} y(t)&\to e^{-2 t} \left (e^t-2\right ) \end{align*}

✓ Sympy. Time used: 0.080 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 5*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 3, Subs(Derivative(y(t), (t, 2)), t, 0): -7, Subs(Derivative(y(t), (t, 3)), t, 0): 15} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = e^{- t} - 2 e^{- 2 t} \]