14.21.7 problem 7

Internal problem ID [2716]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.15, Higher order equations. Excercises page 263
Problem number : 7
Date solved : Sunday, March 30, 2025 at 12:15:34 AM
CAS classification : [[_high_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 25

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

\[ y(t)\to -\frac {t^2}{2}-2 t-e^t (t-3)-3 \]

✓ Sympy. Time used: 0.132 (sec). Leaf size: 22

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

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