8.21.17 problem 17

Internal problem ID [2726]
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 : 17
Date solved : Tuesday, September 30, 2025 at 05:50:17 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 27

ode:=diff(diff(diff(y(t),t),t),t)+diff(diff(y(t),t),t)+diff(y(t),t)+y(t) = t+exp(-t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (t +2 c_3 +1\right ) {\mathrm e}^{-t}}{2}+c_1 \cos \left (t \right )+c_2 \sin \left (t \right )+t -1 \]

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 44

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

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

✓ Sympy. Time used: 0.122 (sec). Leaf size: 24

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

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