8.21.18 problem 18

Internal problem ID [2727]
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 : 18
Date solved : Tuesday, September 30, 2025 at 05:50:17 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+6 y^{\prime \prime }+4 y^{\prime }+y&=t^{3} {\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+4*diff(diff(diff(y(t),t),t),t)+6*diff(diff(y(t),t),t)+4*diff(y(t),t)+y(t) = t^3*exp(-t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t} \left (\frac {1}{840} t^{7}+c_1 +c_2 \,t^{3}+c_3 t +c_4 \,t^{2}\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 41

ode=D[y[t],{t,4}]+4*D[y[t],{t,3}]+6*D[y[t],{t,2}]+4*D[y[t],t]+y[t]==t^3*Exp[-t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{840} e^{-t} \left (t^7+840 c_4 t^3+840 c_3 t^2+840 c_2 t+840 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.305 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**3*exp(-t) + y(t) + 4*Derivative(y(t), t) + 6*Derivative(y(t), (t, 2)) + 4*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} + t \left (C_{2} + t \left (C_{3} + t \left (C_{4} + \frac {t^{4}}{840}\right )\right )\right )\right ) e^{- t} \]